#511710

order	512
nilpotency class	4
isoclinism	isoclinic class of #511571
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 ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

442 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)(27,31)(28,32)(29,33)(30,34)(35,49)(36,44)(37,45)(38,46)(39,47)(40,48)(41,51)(42,52)(43,53)(50,55)(54,56);(1,3)(2,5)(4,7)(6,8)(10,14)(12,19)(15,21)(17,20)(25,27,28,36)(26,31,32,44)(29,35,39,53)(30,42,40,38)(33,49,47,43)(34,52,48,46)(37,55,54,41)(45,50,56,51);(2,6)(5,8)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(26,32)(27,35)(29,41)(30,40)(31,43)(33,50)(36,53)(37,38)(39,55)(42,54)(44,49)(45,52)(46,56)(47,51);(3,7)(5,8)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(27,36)(31,44)(35,53)(37,54)(38,42)(43,49)(45,56)(46,52)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=e,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418906);