#6481888

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1742 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)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,26,29)(23,31,36,48)(24,28,37,45)(25,39,32,30)(27,40,43,35)(33,46,34,47)(38,52,49,44)(41,51,42,50);(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,32,40,26,36,25,35)(22,27,39,48,29,43,30,31)(24,34,49,50,37,33,38,51)(28,42,52,46,45,41,44,47);(2,4)(7,13)(10,17)(14,15)(18,19)(21,24)(22,28)(23,33)(25,38)(26,37)(27,41)(29,45)(30,44)(31,46)(32,49)(34,36)(35,50)(39,52)(40,51)(42,43)(47,48);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(21,24)(22,28)(23,34)(25,38)(26,37)(27,42)(29,45)(30,44)(31,47)(32,49)(33,36)(35,51)(39,52)(40,50)(41,43)(46,48);(6,12)(10,17)(11,20)(18,19)(22,29)(24,37)(27,43)(30,39)(31,48)(33,34)(38,49)(50,51)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^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, 6389084);