#7581724

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

1357 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)(9,10,12,13)(11,15,16,14)(17,18)(19,23)(20,24)(21,26)(22,25)(27,34)(28,35)(29,37)(30,36)(31,38)(32,39)(33,40)(41,45)(42,46)(43,47)(44,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,25,36)(20,27,31,42)(21,28,33,29)(24,34,38,46)(26,35,40,37)(32,43,44,41)(39,47,48,45);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,38)(26,39)(28,43)(29,41)(30,42)(33,44)(35,47)(36,46)(37,45)(40,48);(10,13)(14,15)(19,28)(20,31)(21,33)(23,35)(24,38)(26,40)(27,41)(29,30)(34,45)(36,37)(42,43)(46,47);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(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=i,c^2=fg,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,d^c=di,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, 7488920);