#6467679

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,23)(15,20,28,36)(16,21,30,38)(17,22,29,39)(18,24,32,41)(25,34,31,40)(26,35,43,44)(27,37,33,42);(1,2,3,4)(13,15)(14,20)(16,26)(17,31)(18,33)(19,28)(21,35)(22,40)(23,36)(24,42)(25,29)(27,32)(30,43)(34,39)(37,41)(38,44);(6,9)(10,12)(13,16,18,29,19,30,32,17)(14,21,24,39,23,38,41,22)(15,25,27,26,28,31,33,43)(20,34,37,35,36,40,42,44);(13,17,32,30,19,29,18,16)(14,22,41,38,23,39,24,21)(15,26,33,25,28,43,27,31)(20,35,42,34,36,44,37,40);(5,7,8,11)(6,10,9,12)(14,23)(20,36)(21,38)(22,39)(24,41)(34,40)(35,44)(37,42)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374875);