#152046

order	512
nilpotency class	5
isoclinism	isoclinic class of #149780
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

952 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 82 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)(5,6,9,14,10,15,25,32)(7,16,18,31,19,26,34,35)(8,12,21,29,22,13,24,30)(11,23,27,20,28,33,36,17);(5,7,10,19)(6,11,15,28)(8,20,22,17)(9,23,25,33)(12,16,13,26)(14,31,32,35)(18,24,34,21)(27,29,36,30);(2,4)(5,8,10,22)(6,12,15,13)(7,17,19,20)(9,21,25,24)(11,26,28,16)(14,29,32,30)(18,23,34,33)(27,35,36,31)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=dhi,f^b=fgh,g^b=gi,f^d=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, 59242);