#383221

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×#3010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1877 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=i,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, 290417);