#6675402

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(1,2)(3,4)(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,22,26,30)(23,31,34,42)(24,28,37,46)(25,38,40,29)(27,39,43,33)(32,47,35,49)(36,51,52,45)(41,50,44,48);(2,4)(21,23,26,34)(22,27,30,43)(24,35,37,32)(25,39,40,33)(28,44,46,41)(29,42,38,31)(36,48,52,50)(45,47,51,49);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,26,37)(22,28,30,46)(23,32,34,35)(25,36,40,52)(27,41,43,44)(29,45,38,51)(31,47,42,49)(33,48,39,50);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(22,29)(25,40)(27,42)(28,45)(30,38)(31,43)(33,39)(36,52)(41,49)(44,47)(46,51)(48,50)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582598);