#6486444

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=h,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6393640);