#6500404

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500401
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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2834 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)(13,14,17,20)(15,21,23,28)(16,25,22,19)(18,26,27,24);(5,6)(7,11)(8,10)(9,12)(13,15,22,26,17,23,16,24)(14,18,25,28,20,27,19,21);(2,4)(5,7)(6,9)(8,12)(10,11)(15,23)(18,27)(21,28)(24,26);(7,12)(9,11);(6,10)(9,11)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407600);