#72332374

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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 6045 conjugacy classes of subgroups; only the 3373 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=i,d^2=h,b^a=bj,c^b=ci,d^b=dgh,d^c=dhi,e^b=eg,e^c=eg,e^d=eg,f^a=fi,f^c=fi,f^d=fj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72332391
maximal normal subgroup of	#72332391
maximal quotient of	#72333332

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, 2, F.7);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);