#6673429

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1716 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,27,24)(23,33,36,29)(25,38,40,30)(26,31,28,32)(34,37,46,42)(35,43,50,47)(39,45,41,44)(48,52,49,51);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,27,36)(22,29,24,33)(25,35,40,50)(26,37,28,42)(30,43,38,47)(31,34,32,46)(39,51,41,52)(44,48,45,49);(2,4)(21,24,27,22)(23,34,36,46)(25,30,40,38)(26,31,28,32)(29,42,33,37)(35,49,50,48)(39,45,41,44)(43,51,47,52);(6,11)(10,13)(12,19)(18,20)(21,25,27,40)(22,30,24,38)(23,35,36,50)(26,39,28,41)(29,43,33,47)(31,44,32,45)(34,48,46,49)(37,51,42,52);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,26)(22,31)(23,36)(24,32)(25,39)(27,28)(29,33)(30,44)(35,50)(38,45)(40,41)(43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6580625);