#187628

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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×#3548	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2400 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 2400 conjugacy classes of subgroups — too many to draw. Only its 322 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)(3,4)(5,6,9,14,10,11,8,13)(7,15,16,12,17,20,18,19)(22,26,28,31)(27,35,36,34);(6,11)(8,9)(12,19)(16,18)(21,22)(23,27)(24,31)(25,28)(26,33)(29,34)(30,36)(32,35);(2,4)(5,7,10,17)(6,12,11,19)(8,16,9,18)(13,20,14,15)(21,23,25,30)(22,27,28,36)(24,32,33,29)(26,34,31,35);(5,8,10,9)(6,13,11,14)(7,16,17,18)(12,20,19,15)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh,g^c=gh

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, 94824);