#8806920

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

4917 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,20,24,23)(21,25,30,32)(27,28,29,31);(2,6)(4,7)(9,10,12,14)(11,13,16,15)(17,19,22,24)(18,23,26,20)(21,29,30,27)(25,28,32,31);(2,6)(3,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,24,20)(21,25,30,32)(27,31,29,28);(1,3,5,8)(2,7,6,4)(9,11,12,16)(10,13,14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32);(1,2)(3,7)(4,8)(5,6)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,29,32,27);(1,4,5,7)(2,8,6,3)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714116);