#6497654

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1868 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,32)(24,29)(25,39)(26,30)(27,35)(28,37)(31,40)(33,48)(34,46)(36,47)(38,44)(41,51)(42,52)(43,50)(45,49);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,25,35,26,36,40,52)(22,27,31,32,30,42,39,47)(24,34,38,51,37,33,49,50)(28,43,44,48,29,41,45,46);(1,2,3,4)(7,13)(10,17)(14,19)(18,20)(21,24)(22,28)(23,33)(25,38)(26,37)(27,41)(29,30)(31,44)(32,46)(34,36)(35,50)(39,45)(40,49)(42,43)(47,48)(51,52);(7,13)(10,17)(14,19)(18,20)(21,24)(22,29)(23,34)(25,38)(26,37)(27,41)(28,30)(31,45)(32,46)(33,36)(35,51)(39,44)(40,49)(42,43)(47,48)(50,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,30)(24,37)(27,42)(31,39)(32,47)(33,34)(38,49)(50,51)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

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, 6404850);