#8635465

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

4783 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8542661);