#209190

order	512
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

890 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 204 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 — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=g,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116386);