#403867

order	512
nilpotency class	3
isoclinism	isoclinic class of #396107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=gi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^d=hi

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, 311063);