#491252

order	512
nilpotency class	4
isoclinism	isoclinic class of #491245
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

451 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 256

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

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

Polycyclic presentation:

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

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, 398448);