#6579379

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39356	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2032 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)(7,10)(8,9)(11,12)(13,14,19,21)(15,22,25,23)(16,18,27,17)(20,24,28,26);(1,2)(3,4)(6,9)(10,12)(13,15,18,26,19,25,17,24)(14,20,16,22,21,28,27,23);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,21,19,27,17,14)(15,23,26,20,25,22,24,28);(1,2)(3,4)(13,17,19,18)(14,16,21,27)(15,24,25,26)(20,22,28,23);(2,4)(15,25)(20,28)(22,23)(24,26)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^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=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, 6486575);