#8436552

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4760 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,d^2=gh,e^2=g,f^2=g,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, 8343748);