#6574349

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2731 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481545);