#152932

order	512
nilpotency class	5
isoclinism	isoclinic class of #149911
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

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×#127	C₂×C₄
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

414 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ei,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=dgh,f^b=fgh,g^b=ghi,h^b=hi,f^d=fhi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60128);