#8429318

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	#34925	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5057 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8336514);