#6673865

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1700 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6581061);