#8709018

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

4809 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8616214);