#9232842

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
prime radical	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4333 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9140038);