#511619

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

471 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)(3,5)(4,6)(7,8)(9,10,12,15)(11,14,16,13)(17,18,21,26)(19,28,31,24)(20,34,23,25)(22,27,36,43)(29,44,46,38)(30,39,33,41)(32,45,47,40)(35,48,37,42);(1,3)(2,5)(4,7)(6,8)(9,11,12,16)(10,13,15,14)(17,19,21,31)(18,24,26,28)(20,33,23,30)(22,29,36,46)(25,41,34,39)(27,38,43,44)(32,35,47,37)(40,48,45,42);(2,6)(5,8)(9,11)(10,14)(12,16)(13,15)(19,29)(20,23)(22,37)(24,38)(25,34)(27,42)(28,44)(30,47)(31,46)(32,33)(35,36)(39,40)(41,45)(43,48);(3,7)(5,8)(11,16)(13,14)(17,20,21,23)(18,25,26,34)(19,30,31,33)(22,35,36,37)(24,39,28,41)(27,42,43,48)(29,32,46,47)(38,45,44,40)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=i,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, 418815);