#7124758

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7031954);