#383218

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	C₂×#3008	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1963 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,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, 290414);