#6621342

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	#39131	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1901 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)(13,14)(15,24)(16,29)(17,22)(18,33)(19,23)(20,28)(21,31)(25,40)(26,35)(27,39)(30,37)(32,36)(34,43)(38,42)(41,44);(2,4)(6,10)(8,11)(13,15,18,28,19,27,17,26)(14,20,22,24,23,35,33,39)(16,25,30,41,31,42,32,43)(21,34,37,38,29,44,36,40);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,21)(15,25)(17,32)(18,30)(19,31)(20,34)(22,37)(23,29)(24,38)(26,43)(27,42)(28,41)(33,36)(35,44)(39,40);(5,7,9,12)(6,11,10,8)(13,17,19,18)(14,22,23,33)(15,26,27,28)(16,30,31,32)(20,24,35,39)(21,36,29,37)(25,41,42,43)(34,40,44,38);(5,8,9,11)(6,7,10,12)(15,27)(20,35)(24,39)(25,42)(26,28)(34,44)(38,40)(41,43)

Polycyclic presentation:

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

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, 6528538);