#511701

order	512
nilpotency class	4
isoclinism	isoclinic class of #511577
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418897);