#382972

order	512
nilpotency class	3
isoclinism	isoclinic class of #188766
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

636 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:

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

Polycyclic presentation:

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

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, 290168);