#6782814

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1196 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi

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, 6690010);