#6573981

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2785 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)(11,16)(12,22)(13,19)(14,18)(15,20)(17,23)(21,24);(1,2,3,4)(5,6)(7,8)(9,11,15,12)(10,16,20,22)(13,23,14,21)(17,18,24,19);(6,8)(9,12,15,11)(10,17,20,24)(13,23,14,21)(16,18,22,19);(5,6)(7,8)(9,13,15,14)(10,18,20,19)(11,21,12,23)(16,17,22,24);(6,8)(9,13,15,14)(10,19,20,18)(11,21,12,23)(16,24,22,17)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263965, #72263981, #72264074
maximal normal subgroup of	#72263965, #72263981, #72264074

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481177);