#364168

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271364);