#6675170

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1779 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)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22,26,28)(23,32,33,44)(24,36,37,30)(25,31,27,40)(29,41,42,35)(34,49,48,47)(38,46,39,45)(43,51,52,50);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(22,28)(23,33)(24,37)(31,40)(35,41)(38,39)(43,52)(47,49);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,23)(22,29)(24,34)(25,35)(26,33)(27,41)(28,42)(30,43)(31,44)(32,40)(36,52)(37,48)(38,50)(39,51)(45,47)(46,49);(6,13)(11,15)(12,14)(19,20)(21,24,26,37)(22,30,28,36)(23,34,33,48)(25,39,27,38)(29,43,42,52)(31,46,40,45)(32,47,44,49)(35,51,41,50);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(21,25,26,27)(22,31,28,40)(23,35,33,41)(24,38,37,39)(29,44,42,32)(30,45,36,46)(34,50,48,51)(43,47,52,49)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=i,e^2=f,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, 6582366);