#8107595

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

4757 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=i,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, 8014791);