#492278

order	512
nilpotency class	4
isoclinism	isoclinic class of #492252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=cg,d^b=dei,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399474);