#8097728

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

4801 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)(9,10,13,15)(11,12,16,14)(18,24)(20,25)(26,32)(27,30);(1,3,7,6)(2,5,8,4)(9,11,13,16)(10,14,15,12)(17,18,23,24)(19,25,29,20)(21,26,28,32)(22,27,31,30);(1,4,7,5)(2,3,8,6)(9,11)(10,12)(13,16)(14,15)(17,19,23,29)(18,25,24,20)(21,22,28,31)(26,27,32,30);(1,4,7,5)(2,3,8,6)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,30)(22,26)(23,25)(24,29)(27,28)(31,32);(1,5,7,4)(2,3,8,6)(17,21)(18,26)(19,22)(20,30)(23,28)(24,32)(25,27)(29,31);(1,6,7,3)(2,5,8,4)(17,22)(18,27)(19,28)(20,26)(21,29)(23,31)(24,30)(25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,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, 8004924);