#140112

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×D₈
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

260 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(2,3)(5,6,10,16,12,17,28,15,13,18,9,27,11,8,25,26)(7,14,21,30,23,31,36,29,24,32,20,35,22,19,33,34);(1,2,4,3)(5,7,10,21,12,23,28,36,13,24,9,20,11,22,25,33)(6,14,16,30,17,31,15,29,18,32,27,35,8,19,26,34);(1,3,4,2)(5,8)(6,12)(7,19)(9,15)(10,26)(11,18)(13,17)(14,23)(16,28)(20,29)(21,34)(22,32)(24,31)(25,27)(30,36)(33,35)

Polycyclic presentation:

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

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, 47308);