#6687602

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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1726 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)(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(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)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(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);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(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)(6,13,14,12)(7,16,17,18)(11,20,15,19)(24,35)(28,45)(31,49)(34,50)(36,38)(40,51)(44,52)(46,48);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(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=i,c^2=fg,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, 6594798);