#109815

order	512
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

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	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

598 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=ghi,d^b=di,e^b=eh,d^c=dh,e^c=ehi

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, 17011);