#8689877

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

5730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8597073);