#6598541

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2)(3,4)(13,14)(15,24)(16,27)(17,22)(18,25)(19,23)(20,26)(21,28);(5,6,8,9)(7,10,11,12)(13,15,17,27,19,21,18,26)(14,20,25,28,23,16,22,24);(2,4)(6,9)(7,11)(13,16,18,28,19,20,17,24)(14,21,22,27,23,15,25,26);(5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,21)(17,25)(18,22)(19,23)(24,26)(27,28);(6,9)(7,11)(13,17,19,18)(14,22,23,25)(15,26,21,27)(16,24,20,28)

Polycyclic presentation:

2^9|b^2=fh,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264011, #72264022, #72264050
maximal normal subgroup of	#72264011, #72264022, #72264050
maximal quotient of	#72264917

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505737);