#7584320

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

1277 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)(19,23)(20,24)(21,26)(22,25)(27,34)(28,35)(29,37)(30,36)(31,39)(32,38)(33,40)(41,45)(42,47)(43,46)(44,48);(2,5)(6,8)(9,11,14,15)(10,12,16,13)(17,19,22,30)(18,23,25,36)(20,27,31,42)(21,28,33,29)(24,34,39,47)(26,35,40,37)(32,41,44,43)(38,45,48,46);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,15,16,11)(17,20,22,31)(18,24,25,39)(19,27,30,42)(21,32,33,44)(23,34,36,47)(26,38,40,48)(28,41,29,43)(35,45,37,46);(10,16)(11,15)(19,28)(21,33)(23,35)(26,40)(27,41)(29,30)(32,44)(34,45)(36,37)(38,48)(42,43)(46,47);(2,5)(6,8)(9,13,14,12)(10,15,16,11)(18,25)(20,31)(23,36)(26,40)(27,42)(32,44)(35,37)(41,43)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=fgi,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, 7491516);