#8687570

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

4946 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8594766);