#356580

order	512
nilpotency class	2
isoclinism	isoclinic class of #180873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#376
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

879 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)(3,6)(4,5)(7,8)(9,10)(11,19,23,21)(12,16)(13,17)(14,18)(15,22,29,28)(20,33,36,34)(24,30)(25,31)(26,32)(27,35,39,38)(37,40);(2,5)(3,7)(9,11)(10,15,18,29)(12,20)(13,22)(14,23)(16,27,31,39)(17,19,32,21)(24,35)(25,36)(26,28)(30,33,40,34)(37,38);(2,5)(6,8)(9,12,13,24,14,25,26,37)(10,16,17,30,18,31,32,40)(11,20,22,35,23,36,28,38)(15,27,19,33,29,39,21,34);(1,3,4,7)(2,6,5,8)(9,10)(11,21,23,19)(12,16)(13,17)(14,18)(15,28,29,22)(20,34,36,33)(24,30)(25,31)(26,32)(27,38,39,35)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263776);