#371869

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,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, 279065);