#10409835

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

5148 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)(6,7)(9,10)(11,15)(12,14)(13,16)(17,18,24,22)(19,25,30,29)(20,31,21,26)(23,32,28,27);(1,2,4,5)(3,6,8,7)(9,10,12,14)(11,13,16,15)(17,19,24,30)(18,25,22,29)(20,28,21,23)(26,32,31,27);(9,10)(11,13)(12,14)(15,16)(17,20,24,21)(18,26,22,31)(19,28,30,23)(25,32,29,27);(1,2,4,5)(3,7,8,6)(9,11,12,16)(10,13,14,15)(17,21,24,20)(18,26,22,31)(19,23,30,28)(25,32,29,27);(1,2)(3,6)(4,5)(7,8)(17,22,24,18)(19,29,30,25)(20,31,21,26)(23,32,28,27);(1,3)(2,6)(4,8)(5,7)(17,23)(18,27)(19,20)(21,30)(22,32)(24,28)(25,26)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10317031);