#6583228

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 6490424);