#6571752

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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

2823 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)(21,22,27,32)(23,33,36,46)(24,29,37,43)(25,31,26,30)(28,40,42,35)(34,47,48,52)(38,45,39,44)(41,50,51,49);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,23)(22,28)(24,34)(25,35)(26,40)(27,36)(29,41)(30,33)(31,46)(32,42)(37,48)(38,49)(39,50)(43,51)(44,47)(45,52);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,24)(22,29)(23,34)(25,38)(26,39)(27,37)(28,41)(30,44)(31,45)(32,43)(33,47)(35,49)(36,48)(40,50)(42,51)(46,52);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(24,37)(29,43)(34,48)(38,39)(41,51)(44,45)(47,52)(49,50);(8,16)(11,18)(14,20)(17,19)(21,25,27,26)(22,30,32,31)(23,35,36,40)(24,38,37,39)(28,33,42,46)(29,44,43,45)(34,49,48,50)(41,47,51,52)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478948);