#371870

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1124 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=ef,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^c=hi

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, 279066);