#511713

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

440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,22)(20,24)(25,26,28,31)(27,35,37,32)(29,33,39,44)(30,34,40,45)(36,46,54,47)(38,52,43,48)(41,51,53,49)(42,56,55,50);(1,3)(2,5)(4,7)(6,8)(10,14)(12,19)(15,21)(17,20)(26,31)(27,36)(29,41)(30,40)(32,46)(33,49)(35,47)(37,54)(38,55)(39,53)(42,43)(44,51)(48,50)(52,56);(2,6)(5,8)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,27,28,37)(26,32,31,35)(29,42,39,55)(30,43,40,38)(33,50,44,56)(34,52,45,48)(36,53,54,41)(46,51,47,49);(3,7)(5,8)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(26,31)(27,37)(33,44)(34,45)(36,54)(38,43)(42,55)(49,51)

Polycyclic presentation:

2^9|a^2=f,c^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, 418909);