#511715

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

423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 418911);