#6675729

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

1727 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)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,25,27)(23,31,33,44)(24,35,36,29)(26,39,38,30)(28,40,41,34)(32,46,47,45)(37,43,51,50)(42,48,49,52);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(22,27)(28,41)(29,35)(30,39)(31,44)(42,49)(43,50)(45,46);(2,4)(21,23,25,33)(22,28,27,41)(24,32,36,47)(26,40,38,34)(29,42,35,49)(30,44,39,31)(37,52,51,48)(43,46,50,45);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,25,36)(22,29,27,35)(23,32,33,47)(26,37,38,51)(28,42,41,49)(30,43,39,50)(31,45,44,46)(34,48,40,52);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(24,36)(29,35)(32,47)(37,51)(42,49)(43,50)(45,46)(48,52)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=f,e^2=fi,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, 6582925);