#8912611

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

4921 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)(3,8,4,6)(17,18)(19,25)(20,21)(22,26)(23,29)(24,30)(27,32)(28,31);(3,4)(6,8)(9,10)(11,16)(12,14)(13,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,32)(25,26)(30,31);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,14)(15,16)(20,30)(21,24)(27,31)(28,32);(3,4)(6,8)(11,14)(12,16)(17,20)(18,24)(19,27)(21,26)(22,30)(23,32)(25,28)(29,31);(1,4,5,3)(2,6,7,8)(9,12,13,16)(10,14,15,11)(17,19)(18,25)(20,31)(21,28)(22,29)(23,26)(24,32)(27,30);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(17,21,22,24)(18,20,26,30)(19,28,29,32)(23,27,25,31)

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8819807);