#7586206

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 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)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18)(19,24)(20,26)(21,27)(22,28)(23,29)(25,36)(30,43)(31,40)(32,41)(33,42)(34,39)(35,45)(37,44)(38,46)(47,48);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,33)(18,24,28,42)(20,34,36,30)(21,32,23,38)(25,43,26,39)(27,41,29,46)(31,37,47,35)(40,44,48,45);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,30)(21,37)(22,36)(23,35)(24,39)(26,28)(27,45)(29,44)(31,32)(33,34)(38,47)(40,46)(41,48)(42,43);(9,11,13,12)(10,16,14,15)(17,20)(18,26)(19,31)(21,35)(22,36)(23,37)(24,40)(25,28)(27,45)(29,44)(30,32)(33,47)(34,38)(39,46)(41,43)(42,48);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,28,29)(19,32,33,38)(20,35,36,37)(24,41,42,46)(25,44,26,45)(30,47,34,31)(39,40,43,48)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=gi,e^2=g,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, 7493402);