#72334929

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

69341 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^10|c^2=j,e^2=j,b^a=bj,c^b=ch,d^a=dhi,d^c=dg,e^b=ei,e^d=eghi,f^c=fhi,f^e=fhi

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334931, #72334940
maximal normal subgroup of	#72334931, #72334940
maximal quotient of	#72334980, #72334982, #72335050, …

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