#6583783

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1836 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)(15,22)(16,17)(18,21)(19,28)(20,25)(23,24)(26,27);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,27)(17,24)(18,25)(19,26)(21,22)(23,28);(2,4)(13,16,17,14,18,28,19,21)(15,23,24,22,25,27,26,20);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,16,21,28)(15,24,25,26)(20,27,22,23);(2,4)(15,25)(20,22)(23,27)(24,26)

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490979);