#6675345

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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

1757 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)(28,41)(30,39)(31,44)(32,36)(37,49)(48,51);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,22,25,27)(23,31,33,44)(24,29,35,42)(26,39,38,30)(28,40,41,34)(32,47,36,50)(37,52,49,46)(43,51,45,48);(2,4)(21,23,25,33)(22,28,27,41)(24,36,35,32)(26,40,38,34)(29,45,42,43)(30,44,39,31)(37,48,49,51)(46,47,52,50);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,25,35)(22,29,27,42)(23,32,33,36)(26,37,38,49)(28,43,41,45)(30,46,39,52)(31,47,44,50)(34,48,40,51);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(22,30)(26,38)(27,39)(28,44)(29,46)(31,41)(34,40)(37,49)(42,52)(43,50)(45,47)(48,51)

Polycyclic presentation:

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

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, 6582541);