#7180629

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1325 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)(12,20)(15,21)(19,24)(22,23)(25,26,30,35)(27,31,38,46)(28,39,40,33)(29,32,42,34)(36,49,50,44)(37,43,52,45)(41,48,53,47)(51,55,56,54);(2,5)(3,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,30,38)(26,31,35,46)(28,36,40,50)(29,37,42,52)(32,43,34,45)(33,44,39,49)(41,51,53,56)(47,54,48,55);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(26,32)(29,42)(31,43)(33,47)(34,35)(37,52)(39,48)(41,53)(44,54)(45,46)(49,55)(51,56);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(22,23)(25,28)(26,33)(27,36)(29,41)(30,40)(31,44)(32,47)(34,48)(35,39)(37,51)(38,50)(42,53)(43,54)(45,55)(46,49)(52,56);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(28,40)(33,39)(36,50)(41,53)(44,49)(47,48)(51,56)(54,55)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7087825);