#8096947

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

4817 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=g,e^2=h,f^2=h,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, 8004143);