#7589342

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1537 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)(19,25)(20,32)(21,26)(22,27)(23,28)(24,29)(30,39)(31,40)(33,47)(34,44)(35,41)(36,42)(37,43)(38,45)(46,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,20)(18,25,28,32)(21,35,37,46)(22,31,24,34)(26,41,43,48)(27,40,29,44)(30,38,33,36)(39,45,47,42);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,23,19)(18,25,28,32)(21,33,37,30)(22,31,24,34)(26,39,43,47)(27,44,29,40)(35,36,46,38)(41,45,48,42);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21)(18,26)(19,30)(20,33)(22,36)(23,37)(24,38)(25,39)(27,42)(28,43)(29,45)(31,46)(32,47)(34,35)(40,48)(41,44);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,22,23,24)(18,27,28,29)(19,31,20,34)(21,36,37,38)(25,40,32,44)(26,42,43,45)(30,46,33,35)(39,48,47,41)

Polycyclic presentation:

2^9|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,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, 7496538);