#383220

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1915 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^c=hi,e^d=ei,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, 290416);