#8926171

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

4965 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833367);