#6675188

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1764 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)(21,22,26,30)(23,27,32,41)(24,34,35,44)(25,37,39,29)(28,38,43,36)(31,45,46,52)(33,49,48,42)(40,50,51,47);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(21,23)(22,27)(24,31)(25,33)(26,32)(28,40)(29,42)(30,41)(34,45)(35,46)(36,47)(37,49)(38,50)(39,48)(43,51)(44,52);(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)(26,35)(27,40)(29,44)(30,43)(32,46)(33,50)(34,37)(36,39)(41,51)(42,52)(45,49)(47,48);(6,13)(11,15)(12,14)(19,20)(23,32)(27,41)(31,46)(33,48)(40,51)(42,49)(45,52)(47,50);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(24,35)(28,43)(31,46)(34,44)(36,38)(40,51)(45,52)(47,50)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,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, 6582384);