#7590144

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

1604 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)(17,18,21,25)(19,27,30,23)(20,32,34,24)(22,26,36,42)(28,37,33,40)(29,39,31,38)(35,45,46,41)(43,47,44,48);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,33)(21,30)(22,29)(24,40)(25,27)(26,38)(28,34)(31,36)(32,37)(35,44)(39,42)(41,48)(43,46)(45,47);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,21,34)(18,24,25,32)(19,28,30,33)(22,35,36,46)(23,37,27,40)(26,41,42,45)(29,43,31,44)(38,47,39,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(19,29)(22,36)(23,38)(26,42)(27,39)(28,43)(30,31)(33,44)(35,46)(37,47)(40,48)(41,45);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(18,25)(20,34)(23,27)(26,42)(28,33)(35,46)(38,39)(43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=fgi,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,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, 7497340);