#8757866

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

5290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=g,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, 8665062);