#8891668

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

4819 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8798864);