#8926331

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

5146 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=g,e^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, 8833527);