#6817334

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1637 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgh,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724530);