#129690

order	512
nilpotency class	3
isoclinism	isoclinic class of #1795
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

673 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 673 conjugacy classes of subgroups — too many to draw. Only its 148 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,f^c=fg

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36886);