#6487154

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1231 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,34)(27,37,39,49)(28,42,46,52)(29,36,30,35)(32,41,48,40)(33,44,51,45)(38,47,43,50)(53,54,56,55);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,29,40,31,39,30,41)(26,32,35,37,34,48,36,49)(28,43,44,55,46,38,45,54)(33,50,52,56,51,47,42,53);(11,17)(14,21)(18,19)(22,23)(25,28,29,44,31,46,30,45)(26,33,35,52,34,51,36,42)(27,38,40,54,39,43,41,55)(32,47,37,53,48,50,49,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(27,39)(32,48)(37,49)(38,43)(40,41)(47,50)(53,56)(54,55);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(26,34)(27,39)(33,51)(35,36)(38,43)(40,41)(42,52)(54,55)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=hi,e^2=fg,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, 6394350);