#6674130

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	#42992	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1831 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,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, 6581326);