#6744411

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=fgi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6651607);