#107747

order	512
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

744 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,d^b=di,e^b=ehi,f^b=fi,d^c=dhi,e^c=eh,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14943);