#72326299

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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 10046 conjugacy classes of subgroups; only the 3413 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)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,17,18)(15,19,20,16);(2,4)(5,7)(6,9)(8,12)(10,11)(14,18)(15,20)(21,22)(23,24)(25,28)(26,27);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,17,20)(14,19,18,16)(21,23)(22,26)(24,28)(25,27);(1,2,3,4)(14,18)(15,20)(21,23)(22,24)(25,27)(26,28);(5,6)(7,9)(8,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20)(21,24)(22,27)(23,28)(25,26)

Polycyclic presentation:

2^10|a^2=j,c^2=j,d^2=hij,b^a=bg,c^b=cgh,d^a=dhi,d^b=dhij,d^c=dhi,f^b=fi,f^c=fj,f^d=fi,f^e=fgij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326378
maximal normal subgroup of	#72326378
maximal quotient of	#72333219

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