#6438262

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2738 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:

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

Polycyclic presentation:

2^9|c^2=f,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,e^c=ei,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, 6345458);