#6497365

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1337 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,37)(28,42)(29,34)(30,35)(31,36)(32,39)(33,44)(38,46)(40,47)(41,48)(43,50)(45,49)(51,54)(52,55)(53,56);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,30,41,31,40,29,39)(26,32,34,47,36,48,35,37)(28,38,44,54,45,53,43,52)(33,46,42,55,50,56,49,51);(1,3,4,7)(2,6,5,8)(11,17)(14,21)(18,23)(22,24)(25,28,29,43,31,45,30,44)(26,33,35,49,36,50,34,42)(27,38,39,52,40,53,41,54)(32,46,37,51,48,56,47,55);(11,17)(14,21)(18,23)(22,24)(25,29,31,30)(26,34,36,35)(27,39,40,41)(28,43,45,44)(32,47,48,37)(33,42,50,49)(38,52,53,54)(46,55,56,51);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,40)(32,48)(37,47)(38,53)(39,41)(46,56)(51,55)(52,54)

Polycyclic presentation:

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

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, 6404561);