#8097709

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

4777 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=g,e^2=g,f^2=g,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, 8004905);