#6443398

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1970 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)(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,20)(17,22)(18,21)(23,32)(24,33)(25,35)(26,34)(27,36)(28,38)(29,37)(30,39)(31,40)(41,43)(42,44);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,27)(17,23)(18,25)(20,36)(21,35)(22,32)(24,28)(26,30)(29,42)(31,41)(33,38)(34,39)(37,44)(40,43);(1,2,3,4)(15,23)(17,30)(19,32)(22,39)(24,27)(25,26)(28,31)(33,36)(34,35)(38,40)(41,42)(43,44);(13,16,18,29)(14,20,21,37)(15,24,25,41)(17,31,30,28)(19,33,35,43)(22,40,39,38)(23,27,26,42)(32,36,34,44);(14,21)(15,25)(20,37)(22,39)(23,26)(24,41)(27,42)(38,40)

Polycyclic presentation:

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

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, 6350594);