#6744340

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ 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)(3,7)(4,8)(5,6)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,34)(27,31,38,45)(28,32,39,46)(29,33,41,48)(35,42,37,44)(36,43,51,55)(40,47,52,56)(49,53,50,54);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,31)(28,36)(29,35)(30,38)(32,43)(33,42)(34,45)(37,41)(39,51)(40,49)(44,48)(46,55)(47,53)(50,52)(54,56);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(27,35)(29,41)(31,42)(33,48)(36,49)(37,38)(40,52)(43,53)(44,45)(47,56)(50,51)(54,55);(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)(26,32)(27,36)(29,40)(30,39)(31,43)(33,47)(34,46)(35,49)(37,50)(38,51)(41,52)(42,53)(44,54)(45,55)(48,56);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,35)(28,39)(29,41)(31,42)(32,46)(33,48)(36,50)(37,38)(43,54)(44,45)(49,51)(53,55)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,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, 6651536);