#8692645

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

5409 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=g,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, 8599841);