#8587171

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8494367);