#6802447

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1253 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6709643);