#6677682

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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:

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

Polycyclic presentation:

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

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, 6584878);