#6481970

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,32)(28,39)(29,34)(30,43)(31,35)(33,40)(36,51)(37,46)(38,47)(41,49)(42,48)(44,50)(45,54)(52,56)(53,55);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27)(26,32)(28,36)(29,42)(30,38)(31,37)(33,45)(34,48)(35,46)(39,51)(40,54)(41,52)(43,47)(44,53)(49,56)(50,55);(11,17)(14,21)(18,19)(22,23)(25,28)(26,33)(27,36)(29,40)(30,44)(31,41)(32,45)(34,49)(35,50)(37,52)(38,53)(39,43)(42,54)(46,55)(47,51)(48,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(27,37)(32,46)(36,52)(38,42)(45,55)(47,48)(51,56)(53,54);(10,16)(14,21)(15,24)(22,23)(25,29,31,30)(26,34,35,43)(27,38,37,42)(28,40,41,44)(32,47,46,48)(33,49,50,39)(36,53,52,54)(45,51,55,56)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6389166);