#6686403

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1702 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593599);