#8209857

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

5382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8117053);