#6467906

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

1936 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=hi,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,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, 6375102);