#7589232

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1521 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,28,38)(20,30,33,24)(21,25,36,44)(27,45,46,37)(29,39,34,42)(31,48,47,40)(32,43,35,41);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,23,26,38)(20,31,33,47)(21,34,36,29)(24,40,30,48)(25,42,44,39)(27,35,46,32)(37,43,45,41);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,37)(25,39)(26,38)(30,45)(31,32)(33,46)(34,36)(35,47)(40,41)(42,44)(43,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20,22,33)(18,24,26,30)(19,27,28,46)(21,35,36,32)(23,37,38,45)(25,43,44,41)(29,47,34,31)(39,48,42,40);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,28)(23,38)(27,46)(29,34)(31,47)(37,45)(39,42)(40,48)

Polycyclic presentation:

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

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, 7496428);