#8096108

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

5437 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:

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

Polycyclic presentation:

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

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, 8003304);