#6456667

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2015 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)(13,14,19,24,18,23,17,22)(15,25,26,20,27,28,16,21);(1,2)(3,4)(13,15,18,27)(14,20,23,21)(16,17,26,19)(22,28,24,25);(2,4)(6,9)(10,12)(13,14,17,22,18,23,19,24)(15,25,16,21,27,28,26,20);(2,4)(13,16,19,15,18,26,17,27)(14,21,24,25,23,20,22,28);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,23,24)(15,26,27,16)(20,28,21,25)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,d^2=fi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363863);