#8970599

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

5616 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)(3,7)(4,5)(6,8)(10,12)(14,16)(17,18,23,26)(19,27,29,24)(20,30,31,21)(22,25,28,32);(3,8)(6,7)(11,15)(14,16)(17,19,23,29)(18,24,26,27)(20,25,31,32)(21,28,30,22);(1,3,5,8)(2,6,4,7)(9,10,13,12)(11,14,15,16)(20,31)(21,30)(22,28)(25,32);(3,8)(6,7)(17,20,23,31)(18,21,26,30)(19,25,29,32)(22,24,28,27);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,21)(18,20)(19,28)(22,29)(23,30)(24,25)(26,31)(27,32);(1,4,5,2)(3,6,8,7)(9,12,13,10)(11,16,15,14)(17,22,23,28)(18,25,26,32)(19,21,29,30)(20,24,31,27)

Polycyclic presentation:

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

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, 8877795);