#9223830

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	#37653	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4683 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=hi,e^2=h,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, 9131026);