#6578089

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1992 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)(7,11)(8,12)(9,10)(13,14)(15,21)(16,19)(17,26)(18,24)(20,25)(22,27)(23,28);(1,2)(3,4)(6,10)(11,12)(13,14,16,19)(15,22,23,18)(17,20,25,26)(21,24,28,27);(2,4)(13,15,16,23)(14,18,19,22)(17,27,25,24)(20,28,26,21);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,14)(15,22)(16,19)(17,20)(18,23)(21,27)(24,28)(25,26);(2,4)(5,8,9,7)(6,12,10,11)(14,19)(18,22)(20,26)(21,28)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485285);