#7584322

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1270 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=fgi,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,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, 7491518);