#8433731

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

5110 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8340927);