#7598883

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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

1343 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fh,d^2=fg,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,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, 7506079);