#384295

order	512
nilpotency class	3
isoclinism	isoclinic class of #32696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1005 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi,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, 291491);