#6454186

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1968 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)(20,27)(26,43)(28,42)(29,39)(30,38)(31,37)(32,40)(33,41)(34,36)(35,44);(2,4)(13,15)(14,20)(16,29)(17,31)(18,27)(19,34)(21,37)(22,39)(23,25)(24,42)(26,30)(28,33)(32,44)(35,38)(36,41)(40,43);(6,9)(10,12)(13,16,18,17)(14,21,23,22)(15,26,27,44)(19,32,33,30)(20,35,25,43)(24,40,41,38)(28,31,34,29)(36,39,42,37);(13,17,18,16)(14,22,23,21)(15,26,27,44)(19,30,33,32)(20,35,25,43)(24,38,41,40)(28,31,34,29)(36,39,42,37);(5,7,8,11)(6,10,9,12)(14,23)(20,25)(21,22)(24,41)(35,43)(36,42)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=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, 6361382);