#6486329

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 6393525);