#6675169

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

1786 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)(21,22,25,30)(23,32,34,27)(24,36,38,29)(26,28,40,31)(33,42,37,45)(35,43,48,41)(39,46,50,44)(47,51,49,52);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(21,23)(22,27)(24,37)(25,34)(26,35)(28,41)(29,45)(30,32)(31,43)(33,38)(36,42)(39,49)(40,48)(44,52)(46,51)(47,50);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(22,28)(26,40)(27,41)(29,44)(30,31)(32,43)(35,48)(36,46)(39,50)(42,51)(45,52)(47,49);(6,13)(11,15)(12,14)(19,20)(21,24)(22,29)(23,33)(25,38)(26,39)(27,42)(28,44)(30,36)(31,46)(32,45)(34,37)(35,47)(40,50)(41,51)(43,52)(48,49);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(23,34)(24,38)(27,32)(29,36)(35,48)(39,50)(41,43)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,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, 6582365);