#7181786

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1323 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=fi,d^2=f,e^2=hi,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,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, 7088982);