#8433794

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

5550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(2,3)(6,7)(9,10,13,16)(11,12,15,14)(17,18,20,22)(19,21,23,24);(1,2,5,3)(4,6,8,7)(9,11,13,15)(10,12,16,14)(19,23)(21,24);(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24);(1,2,5,3)(4,7,8,6)(9,11)(10,12)(13,15)(14,16)(17,19)(18,21)(20,23)(22,24);(1,3,5,2)(4,7,8,6)(17,18)(19,21)(20,22)(23,24);(1,4,5,8)(2,6,3,7)(9,12)(10,15)(11,16)(13,14)

Polycyclic presentation:

2^9|a^2=h,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, 8340990);