#6986624

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1324 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)(4,6)(5,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,31,36)(27,37,40,50)(28,33,41,47)(29,35,30,34)(32,44,46,39)(38,51,52,56)(42,49,43,48)(45,53,55,54);(1,3,4,8)(2,5,6,7)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(28,41)(33,47)(38,52)(42,43)(45,55)(48,49)(51,56)(53,54);(3,8)(5,7)(9,10)(11,15)(12,21)(13,16)(14,17)(18,20)(19,24)(22,23)(25,27,31,40)(26,32,36,46)(28,38,41,52)(29,39,30,44)(33,45,47,55)(34,37,35,50)(42,53,43,54)(48,56,49,51);(2,6)(5,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28)(26,33)(27,38)(29,43)(30,42)(31,41)(32,45)(34,49)(35,48)(36,47)(37,51)(39,54)(40,52)(44,53)(46,55)(50,56);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,29,31,30)(26,34,36,35)(27,39,40,44)(28,42,41,43)(32,37,46,50)(33,48,47,49)(38,53,52,54)(45,56,55,51)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6893820);