#6677352

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=f,e^2=f,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, 6584548);