#8525414

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

5357 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8432610);