#8185337

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

5539 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8092533);