#9224207

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	#37808	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gi,e^2=h,f^2=gh,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, 9131403);