#152772

order	512
nilpotency class	5
isoclinism	isoclinic class of #149919
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊Q₈	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 508 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dghi,f^b=fgh,g^b=ghi,h^b=hi,f^d=fhi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59968);