#7539244

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1533 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=g,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,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, 7446440);