#8724858

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

5257 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8632054);