#6675563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

1743 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)(22,27)(24,35)(28,40)(30,38)(31,43)(32,47)(36,49)(48,50);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,22,25,27)(23,31,33,43)(24,29,35,41)(26,38,37,30)(28,39,40,34)(32,46,47,52)(36,44,49,45)(42,50,51,48);(2,4)(21,23,25,33)(22,28,27,40)(24,32,35,47)(26,39,37,34)(29,42,41,51)(30,43,38,31)(36,50,49,48)(44,46,45,52);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24)(22,29)(23,32)(25,35)(26,36)(27,41)(28,42)(30,45)(31,46)(33,47)(34,48)(37,49)(38,44)(39,50)(40,51)(43,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(22,30)(24,35)(26,37)(27,38)(28,43)(29,44)(31,40)(32,47)(34,39)(41,45)(42,46)(51,52)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6582759);