#8757797

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

4893 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:

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

Polycyclic presentation:

2^9|b^2=h,c^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, 8664993);