#8107485

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

4863 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014681);