#8107684

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

5163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014880);