#152224

order	512
nilpotency class	5
isoclinism	isoclinic class of #149771
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

397 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,e^c=eh,f^c=fghi,g^c=gi,f^e=fhi,g^f=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59420);