#6487239

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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

1226 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)(25,26,32,36)(27,39,42,34)(28,43,41,33)(29,35,46,54)(30,38,31,37)(40,55,45,50)(44,51,56,49)(47,53,48,52);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,28,32,42,30,41)(26,33,38,34,36,43,37,39)(29,45,48,56,46,40,47,44)(35,51,53,55,54,49,52,50);(11,17)(14,21)(18,19)(22,23)(25,28,31,42,32,41,30,27)(26,34,38,43,36,39,37,33)(29,44,48,45,46,56,47,40)(35,50,53,51,54,55,52,49);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,32,46)(26,35,36,54)(27,40,42,45)(28,44,41,56)(30,47,31,48)(33,49,43,51)(34,50,39,55)(37,52,38,53);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(26,36)(29,46)(33,43)(34,39)(37,38)(40,45)(44,56)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=h,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^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, 6394435);