#6675190

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1787 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)(21,22,27,31)(23,28,34,44)(24,35,37,45)(25,30,26,39)(29,40,46,36)(32,47,49,52)(33,42,38,43)(41,48,51,50);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(21,23)(22,28)(24,32)(25,38)(26,33)(27,34)(29,41)(30,43)(31,44)(35,47)(36,50)(37,49)(39,42)(40,48)(45,52)(46,51);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24)(22,29)(23,32)(25,36)(26,40)(27,37)(28,41)(30,45)(31,46)(33,48)(34,49)(35,39)(38,50)(42,47)(43,52)(44,51);(6,13)(11,15)(12,14)(19,20)(21,25,27,26)(22,30,31,39)(23,33,34,38)(24,36,37,40)(28,42,44,43)(29,45,46,35)(32,48,49,50)(41,47,51,52);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(24,37)(29,46)(32,49)(35,45)(36,40)(41,51)(47,52)(48,50)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,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,e^c=ei,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582386);