#72335531

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 6556 conjugacy classes of subgroups; only the 3449 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,16)(15,19,17,20)(21,22,27,26)(23,24,25,28);(1,2,4,3)(5,6,9,7)(8,10,12,11)(13,15,18,17)(14,19,16,20)(21,23)(22,24)(25,27)(26,28);(2,3)(6,7)(10,11)(14,16)(19,20)(21,24)(22,25)(23,26)(27,28);(2,3)(6,7)(8,12)(14,16)(15,17)(22,26)(23,25);(1,3,4,2)(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,20,17,19)(21,25,27,23)(22,28,26,24);(5,8,9,12)(6,10,7,11)(13,17,18,15)(14,20,16,19)(21,26,27,22)(23,28,25,24)

Polycyclic presentation:

2^10|a^2=h,b^2=hj,e^2=gh,f^2=gij,b^a=bg,c^a=chj,c^b=chi,d^a=dgh,d^b=dgh,e^a=eg,e^b=egj,e^c=ehj,e^d=egh,f^d=fgij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72335547
maximal normal subgroup of	#72335547
maximal quotient of	#72336109

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 2, F.8*F.10);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetPower(coll, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);