#363092

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

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×#382
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2285 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

773 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)(3,7)(4,6)(5,8)(9,10)(11,21,24,16)(12,25,29,39)(13,30,31,18)(14,20)(15,19)(17,28,36,37)(22,32)(23,38)(26,35)(27,33)(34,40);(1,3,4,8)(2,5,6,7)(9,11,22,31,15,24,14,13)(10,16,32,30,19,21,20,18)(12,26,37,27,29,23,28,40)(17,34,39,35,36,33,25,38);(3,8)(5,7)(9,12,15,29)(10,17)(11,23,24,26)(13,27,31,40)(14,28,22,37)(16,33)(18,35)(19,36)(20,25)(21,34)(30,38)(32,39);(2,6)(5,7)(9,13,14,24,15,31,22,11)(10,18,20,21,19,30,32,16)(12,27,28,26,29,40,37,23)(17,35,25,34,36,38,39,33)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=eg,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,h^c=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, 270288);