#7180391

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1185 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)(9,10,14,17)(11,18,12,15)(13,21,22,16)(19,23,20,24)(25,26)(27,31)(28,33)(29,32)(30,34)(35,42)(36,44)(37,43)(38,45)(39,48)(40,50)(41,47)(46,52)(49,54)(51,55)(53,56);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,14)(13,20)(16,24)(17,18)(19,22)(21,23)(25,27,30,38)(26,31,34,45)(28,39,40,36)(29,37,42,52)(32,43,35,46)(33,48,50,44)(41,53,54,51)(47,56,49,55);(1,4,5,2)(3,7,8,6)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(26,32)(29,42)(31,43)(33,47)(34,35)(37,52)(41,54)(44,55)(45,46)(48,56)(49,50)(51,53);(3,8)(6,7)(11,12)(15,18)(19,20)(23,24)(25,28,30,40)(26,33,34,50)(27,36,38,39)(29,41,42,54)(31,44,45,48)(32,47,35,49)(37,51,52,53)(43,55,46,56);(9,13,14,22)(10,16,17,21)(11,19,12,20)(15,23,18,24)(26,34)(28,40)(31,45)(32,35)(36,39)(41,54)(43,46)(51,53)

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=f,d^2=i,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, 7087587);