#6544256

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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2760 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)(21,22,27,31,26,32,25,30)(23,33,35,28,37,42,36,43)(24,29,38,44,40,46,39,45)(34,47,49,41,50,52,48,51);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23)(22,28)(24,34)(25,35)(26,37)(27,36)(29,41)(30,42)(31,33)(32,43)(38,48)(39,49)(40,50)(44,47)(45,52)(46,51);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24)(22,29)(23,34)(25,39)(26,40)(27,38)(28,41)(30,45)(31,44)(32,46)(33,47)(35,49)(36,48)(37,50)(42,52)(43,51);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,26,27)(22,30,32,31)(23,35,37,36)(24,38,40,39)(28,42,43,33)(29,44,46,45)(34,48,50,49)(41,47,51,52);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,26,27)(22,31,32,30)(23,35,37,36)(24,39,40,38)(28,33,43,42)(29,44,46,45)(34,49,50,48)(41,47,51,52)

Polycyclic presentation:

2^9|a^2=fgi,d^2=hi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6451452);