#9453091

order	512
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	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(26,27)(31,33)(32,34)(38,39);(2,5)(6,7)(12,19)(15,17)(20,24)(22,23)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,2)(3,6)(4,5)(7,8)(10,11)(15,17)(16,18)(22,23)(29,37)(32,34)(35,40)(38,39);(1,3,4,8)(2,6,5,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219281, #72219285
maximal quotient of	#72219273

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9360287);