#152234

order	512
nilpotency class	5
isoclinism	isoclinic class of #149780
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×C₄⋊Q₈	C₂³
prime radical	C₂³	C₂×#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

378 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 70 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)(5,6,8,13,10,15,22,32)(7,16,18,30,17,28,33,36)(9,14,24,31,25,12,21,29)(11,20,27,23,26,34,35,19);(1,2,3,4)(7,17)(11,26)(16,28)(18,33)(19,23)(20,34)(27,35)(30,36);(5,7,10,17)(6,11,15,26)(8,20,22,34)(9,23,25,19)(12,28,14,16)(13,30,32,36)(18,21,33,24)(27,31,35,29)

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,e^c=ehi,f^c=fgh,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, 59430);