#6572684

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3565	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2757 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)(21,22,27,39,26,31,25,30)(23,32,34,28,36,42,35,43)(24,29,40,50,37,44,38,45)(33,46,48,41,47,52,49,51);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,23,26,36)(22,28,31,43)(24,33,37,47)(25,34,27,35)(29,41,44,51)(30,42,39,32)(38,48,40,49)(45,52,50,46);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,24)(22,29)(23,33)(25,38)(26,37)(27,40)(28,41)(30,45)(31,44)(32,46)(34,48)(35,49)(36,47)(39,50)(42,52)(43,51);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(24,37)(29,44)(33,47)(38,40)(41,51)(45,50)(46,52)(48,49);(8,16)(11,18)(14,20)(17,19)(21,25,26,27)(22,30,31,39)(23,34,36,35)(24,38,37,40)(28,42,43,32)(29,45,44,50)(33,48,47,49)(41,52,51,46)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,d^c=di,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6479880);