#6674696

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	#42643	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1813 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)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22,26,30)(23,27,34,43)(24,28,36,45)(25,29,39,48)(31,40,49,51)(32,41,33,42)(35,44,50,52)(37,46,38,47);(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(21,23)(22,27)(24,35)(25,32)(26,34)(28,44)(29,41)(30,43)(31,37)(33,39)(36,50)(38,49)(40,46)(42,48)(45,52)(47,51);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24)(22,28)(23,31)(25,38)(26,36)(27,40)(29,47)(30,45)(32,35)(33,50)(34,49)(37,39)(41,44)(42,52)(43,51)(46,48);(6,13)(11,15)(12,14)(19,20)(24,36)(28,45)(31,49)(35,50)(37,38)(40,51)(44,52)(46,47);(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(23,32)(25,39)(27,41)(29,48)(31,35)(33,34)(37,38)(40,44)(42,43)(46,47)(49,50)(51,52)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=f,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, 6581892);