#202872

order	512
nilpotency class	3
isoclinism	isoclinic class of #19420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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

802 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)(6,10)(7,13)(8,16)(11,19)(12,18)(14,17)(21,22)(23,27)(24,29)(25,34)(26,30)(28,31)(32,36)(33,35);(1,3,4,2)(5,6,8,10,9,12,16,18)(7,14,15,19,13,11,20,17)(21,23)(22,27)(24,33)(25,31)(26,32)(28,34)(29,35)(30,36);(5,7,8,15,9,13,16,20)(6,11,10,17,12,14,18,19)(21,23)(22,28)(24,31)(25,33)(26,32)(27,34)(29,36)(30,35);(6,12)(10,18)(11,14)(17,19)(21,24,26,25)(22,29,30,34)(23,31,32,33)(27,28,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110068);