#386201

order	512
nilpotency class	3
isoclinism	isoclinic class of #220210
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1127 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,7)(6,8)(9,10,16,22,14,20,15,21)(11,23,26,35,25,17,12,18)(13,28,31,38,30,19,32,37)(24,33,27,40,29,36,39,34);(1,3)(2,4)(5,8)(6,7)(9,11)(10,17)(12,15)(13,29)(14,25)(16,26)(18,22)(19,36)(20,23)(21,35)(24,30)(27,32)(28,33)(31,39)(34,37)(38,40);(1,4,6,5)(2,7,8,3)(9,12,16,25,14,26,15,11)(10,18,22,17,20,35,21,23)(13,27,32,29,30,39,31,24)(19,34,38,33,28,40,37,36);(1,5,6,4)(2,7,8,3)(9,13)(10,19)(11,24)(12,27)(14,30)(15,31)(16,32)(17,33)(18,34)(20,28)(21,37)(22,38)(23,36)(25,29)(26,39)(35,40)

Polycyclic presentation:

2^9|a^2=h,c^2=eg,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^c=hi,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, 293397);