#8445590

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

4783 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=g,e^2=i,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, 8352786);