#6818746

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 6725942);