#6486362

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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

1328 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)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,36,32,38,30,37)(27,39,42,33,44,50,43,51)(28,45,46,52,29,35,47,34)(40,53,55,56,41,48,54,49);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27)(26,33)(28,40)(29,41)(30,42)(31,43)(32,44)(34,48)(35,49)(36,39)(37,50)(38,51)(45,56)(46,54)(47,55)(52,53);(11,17)(14,21)(18,19)(22,23)(25,28,32,29)(26,34,38,52)(27,40,44,41)(30,46,31,47)(33,48,51,53)(35,37,45,36)(39,49,50,56)(42,54,43,55);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,32,28)(26,35,38,45)(27,41,44,40)(30,46,31,47)(33,49,51,56)(34,37,52,36)(39,48,50,53)(42,54,43,55);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(25,30,32,31)(26,36,38,37)(27,42,44,43)(28,46,29,47)(33,39,51,50)(34,35,52,45)(40,54,41,55)(48,49,53,56)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6393558);