#9210366

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	#42222	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4431 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9117562);