#7596447

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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×#496	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,34)(21,40)(22,30)(23,29)(24,28)(25,33)(26,39)(31,47)(32,42)(35,44)(36,41)(37,46)(38,45)(43,48);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,33)(18,25,29,27)(20,35,38,36)(21,22,32,24)(26,43,46,47)(28,42,30,40)(31,37,48,39)(34,41,45,44);(10,13)(14,15)(17,20,24,39,23,38,22,37)(18,26,30,34,29,46,28,45)(19,31,32,35,33,48,21,36)(25,41,42,43,27,44,40,47);(9,11,12,16)(10,14,13,15)(17,21,24,19,23,32,22,33)(18,27,30,40,29,25,28,42)(20,36,39,31,38,35,37,48)(26,44,34,47,46,41,45,43);(1,3,4,8)(2,6,5,7)(17,22,23,24)(18,28,29,30)(19,32,33,21)(20,37,38,39)(25,42,27,40)(26,45,46,34)(31,35,48,36)(41,43,44,47)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7503643);