#6482601

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 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)(27,32,41,50)(28,34,44,54)(29,36,30,35)(33,45,52,43)(38,47,42,53)(39,49,40,48)(46,56,51,55);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,41)(26,32,37,50)(28,42,44,38)(29,39,30,40)(33,51,52,46)(34,53,54,47)(35,48,36,49)(43,56,45,55);(11,17)(14,21)(18,19)(22,23)(25,28,31,44)(26,33,37,52)(27,38,41,42)(29,43,30,45)(32,46,50,51)(34,36,54,35)(39,55,40,56)(47,49,53,48);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28)(26,34)(27,38)(29,45)(30,43)(31,44)(32,47)(33,35)(36,52)(37,54)(39,56)(40,55)(41,42)(46,48)(49,51)(50,53);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(25,29,31,30)(26,35,37,36)(27,39,41,40)(28,43,44,45)(32,48,50,49)(33,34,52,54)(38,55,42,56)(46,47,51,53)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,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, 6389797);