#473905

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

934 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=i,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,g^c=gi,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, 381101);