#6640455

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1230 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26,30,35)(27,34,29,32)(28,39,42,55)(31,45,41,36)(33,48,50,43)(37,52,40,54)(38,47,44,51)(46,53,49,56);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,27,30,29)(26,32,35,34)(28,40,42,37)(31,38,41,44)(33,49,50,46)(36,47,45,51)(39,52,55,54)(43,56,48,53);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,41,48,30,42,31,43)(26,33,45,55,35,50,36,39)(27,37,44,56,29,40,38,53)(32,46,51,52,34,49,47,54);(3,7)(5,8)(10,17)(11,20)(16,22)(19,21)(25,29,30,27)(26,34,35,32)(28,37,42,40)(31,44,41,38)(33,46,50,49)(36,51,45,47)(39,54,55,52)(43,53,48,56);(2,6)(5,8)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(27,29)(32,34)(37,40)(38,44)(46,49)(47,51)(52,54)(53,56)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6547651);