#363723

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1135 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270919);