#202755

order	512
nilpotency class	3
isoclinism	isoclinic class of #19230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

792 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109951);