#371574

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	C₄×#121
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1049 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)(11,21,24,32)(12,25)(13,16)(14,19)(15,20)(17,23,33,29)(18,27)(22,31,36,40)(26,30)(28,34)(35,39,38,37);(1,3,4,8)(2,5,6,7)(10,16,20,19)(12,26)(17,21,33,32)(18,25,34,30)(22,36)(27,28)(31,38,40,35)(37,39);(3,8)(5,7)(9,11)(10,17)(12,22)(13,23)(14,29)(15,24)(16,21)(18,31)(19,32)(20,33)(25,38)(26,36)(27,37)(28,39)(30,35)(34,40);(2,6)(5,7)(9,12,13,27,15,26,14,28)(10,18,19,25,20,34,16,30)(11,22,23,37,24,36,29,39)(17,31,32,38,33,40,21,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278770);