#152966

order	512
nilpotency class	5
isoclinism	isoclinic class of #149927
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂×#496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2326 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

370 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 64 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)(7,16,17,31)(8,13,11,26)(9,22,25,32)(10,15)(12,18,30,35)(14,24)(19,33,20,34)(21,27)(23,29,36,28);(5,7,9,23,10,20,25,35)(6,12,27,33,15,29,14,31)(8,17,24,36,11,19,21,18)(13,28,22,34,26,30,32,16);(1,2,3,4)(5,8,10,11)(6,13,15,26)(7,17,20,19)(9,24,25,21)(12,28,29,30)(14,32,27,22)(16,33,34,31)(18,23,36,35)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=eh,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dghi,e^b=ei,f^b=fghi,g^b=ghi,h^b=hi,d^c=di,f^c=fi,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, 60162);