#6583781

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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
prime radical	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263894, #72263916, #72263954
maximal normal subgroup of	#72263894, #72263916, #72263954
maximal quotient of	#72264557

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490977);