#374299

order	512
nilpotency class	3
isoclinism	isoclinic class of #196266
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1541 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)(4,5)(6,8)(9,10)(11,15)(12,22,25,34)(13,17)(14,18)(16,24,32,27)(19,36,23,39)(20,30)(21,31)(26,33)(28,35)(29,37,40,38);(2,5)(6,7)(9,11,14,21)(10,15,18,31)(12,23,25,19)(13,26,28,20)(16,29,32,40)(17,33,35,30)(22,39,34,36)(24,37,27,38);(9,12,13,27,14,25,28,24)(10,16,17,34,18,32,35,22)(11,19,20,38,21,23,26,37)(15,29,30,39,31,40,33,36);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,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, 281495);