#7181875

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

1316 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)(25,26,29,34)(27,36,38,31)(28,33,40,47)(30,32,42,35)(37,49,50,44)(39,45,52,43)(41,46,53,48)(51,55,56,54);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,29,38)(26,31,34,36)(28,37,40,50)(30,39,42,52)(32,43,35,45)(33,44,47,49)(41,51,53,56)(46,54,48,55);(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)(30,42)(31,43)(33,46)(34,35)(36,45)(39,52)(41,53)(44,54)(47,48)(49,55)(51,56);(1,4,5,2)(3,6,8,7)(11,18)(14,17)(19,24)(22,23)(25,28)(26,33)(27,37)(29,40)(30,41)(31,44)(32,46)(34,47)(35,48)(36,49)(38,50)(39,51)(42,53)(43,54)(45,55)(52,56);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(26,34)(28,40)(31,36)(32,35)(37,50)(41,53)(43,45)(51,56)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=fi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,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, 7089071);