#7589965

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=fg,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497161);