#6744630

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1213 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)(3,7)(4,8)(5,6)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,31,36)(27,32,40,47)(28,33,41,48)(29,34,30,35)(37,44,39,46)(38,45,53,56)(42,49,43,50)(51,54,52,55);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,32)(28,38)(29,39)(30,37)(31,40)(33,45)(34,46)(35,44)(36,47)(41,53)(42,52)(43,51)(48,56)(49,55)(50,54);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(27,37)(28,41)(29,30)(32,44)(33,48)(34,35)(38,52)(39,40)(45,55)(46,47)(51,53)(54,56);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28,31,41)(26,33,36,48)(27,38,40,53)(29,42,30,43)(32,45,47,56)(34,49,35,50)(37,51,39,52)(44,54,46,55);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29)(26,34)(28,42)(30,31)(33,49)(35,36)(37,39)(41,43)(44,46)(48,50)(51,52)(54,55)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6651826);