#8459120

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	#39386	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4845 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8366316);