#7620397

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1362 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)(10,14)(11,12)(17,18)(19,29)(20,25)(21,36)(22,27)(23,28)(24,31)(26,32)(30,43)(33,46)(34,40)(35,41)(37,42)(38,44)(39,45)(47,48);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,31)(18,24,27,29)(20,30,34,44)(21,37,32,23)(25,38,40,43)(26,42,36,28)(33,47,45,35)(39,48,46,41);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,22,34)(18,25,27,40)(19,30,31,44)(21,33,32,45)(23,35,37,47)(24,38,29,43)(26,39,36,46)(28,41,42,48);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(17,21,23,19,22,32,37,31)(18,26,28,24,27,36,42,29)(20,33,35,30,34,45,47,44)(25,39,41,38,40,46,48,43);(2,6)(7,8)(9,12,13,11)(10,15,14,16)(18,27)(20,34)(24,29)(26,36)(28,42)(30,44)(33,45)(35,47)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=fghi,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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527593);