#6728861

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

1173 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)(4,7)(5,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26)(27,36)(28,41)(29,45)(30,35)(31,34)(32,40)(33,44)(37,47)(38,48)(39,49)(42,50)(43,51)(46,52)(53,55)(54,56);(1,3,5,4)(2,6,8,7)(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(25,27,30,40)(26,32,35,36)(28,42,44,54)(29,38,31,39)(33,50,41,56)(34,48,45,49)(37,46,55,43)(47,51,53,52);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28)(26,33)(27,37)(29,43)(30,44)(31,46)(32,47)(34,51)(35,41)(36,53)(38,54)(39,42)(40,55)(45,52)(48,50)(49,56);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,30,31)(26,34,35,45)(27,38,40,39)(28,43,44,46)(32,48,36,49)(33,51,41,52)(37,54,55,42)(47,50,53,56);(1,4,5,3)(2,7,8,6)(11,21)(15,18)(19,20)(23,24)(27,39)(29,31)(32,48)(34,45)(36,49)(37,42)(38,40)(43,46)(47,50)(51,52)(53,56)(54,55)

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=fg,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6636057);