#6710066

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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

1211 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)(4,5)(7,8)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,31)(28,39)(29,33)(30,34)(32,41)(35,53)(36,51)(37,47)(38,46)(40,56)(42,52)(43,50)(44,49)(45,54)(48,55);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,29,37)(26,31,33,47)(28,40,41,55)(30,36,43,38)(32,48,39,56)(34,51,50,46)(35,44,54,42)(45,52,53,49);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,29,41)(26,32,33,39)(27,35,37,54)(30,44,43,42)(31,45,47,53)(34,52,50,49)(36,40,38,55)(46,56,51,48);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(28,41)(32,39)(35,54)(40,55)(42,44)(45,53)(48,56)(49,52);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(26,33)(27,36)(28,41)(30,43)(31,46)(35,55)(37,38)(40,54)(45,48)(47,51)(49,52)(53,56)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=fgi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,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, 6617262);