#6583703

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1773 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)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,22)(17,21)(18,23)(20,30)(24,43)(25,35)(26,34)(27,36)(28,41)(29,38)(31,40)(32,42)(33,39)(37,44);(6,9)(10,12)(13,15,17,25)(14,19,21,35)(16,28,30,44)(18,32,31,27)(20,37,22,41)(23,42,40,36)(24,33,26,29)(34,38,43,39);(1,2,3,4)(13,16,17,30)(14,20,21,22)(15,24,25,26)(18,33,31,29)(19,34,35,43)(23,38,40,39)(27,44,32,28)(36,41,42,37);(5,7,8,11)(6,10,9,12)(14,21)(15,25)(16,29)(18,31)(20,38)(22,39)(24,44)(26,28)(30,33)(34,37)(36,42)(41,43);(13,16,17,30)(14,22,21,20)(15,26,25,24)(18,33,31,29)(19,34,35,43)(23,39,40,38)(27,28,32,44)(36,41,42,37)

Polycyclic presentation:

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

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, 6490899);