#8093295

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

4894 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)(6,8)(10,12)(14,15)(18,20)(22,24)(25,28)(29,31);(2,5)(3,7)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,24,27,22)(21,28,30,25)(26,29,32,31);(1,2,4,5)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,19,23,27)(18,24,20,22)(21,26,30,32)(25,31,28,29);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20,23,18)(19,22,27,24)(21,28,30,25)(26,29,32,31);(17,21,23,30)(18,25,20,28)(19,26,27,32)(22,29,24,31);(17,22)(18,19)(20,27)(21,29)(23,24)(25,26)(28,32)(30,31)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000491);