#6776320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6683516);