#6479214

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1352 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=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, 6386410);