#7589361

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1539 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)(17,18,23,29)(19,26,20,25)(21,27,36,45)(22,37,24,28)(30,42,32,39)(31,40,33,41)(34,48,47,43)(35,46,38,44);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,20)(18,25,29,26)(21,34,36,47)(22,31,24,33)(27,43,45,48)(28,40,37,41)(30,38,32,35)(39,44,42,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,23,19)(18,26,29,25)(21,32,36,30)(22,31,24,33)(27,42,45,39)(28,40,37,41)(34,35,47,38)(43,46,48,44);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21)(18,27)(19,30)(20,32)(22,35)(23,36)(24,38)(25,39)(26,42)(28,44)(29,45)(31,47)(33,34)(37,46)(40,43)(41,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,22,23,24)(18,28,29,37)(19,31,20,33)(21,35,36,38)(25,40,26,41)(27,44,45,46)(30,47,32,34)(39,43,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=fg,e^2=fgi,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, 7496557);