#7040193

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

1187 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,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, 6947389);