#491686

order	512
nilpotency class	4
isoclinism	isoclinic class of #491653
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

559 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egi,c^2=fh,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=ci,d^c=dg

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, 398882);