#374286

order	512
nilpotency class	3
isoclinism	isoclinic class of #196110
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

1515 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)(12,16,23,33)(13,18)(14,17)(19,28,37,39)(20,30)(21,29)(22,31,38,40)(24,32,26,36)(25,35)(27,34);(2,5)(6,7)(9,11,14,21)(10,15,17,29)(12,22,23,38)(13,25,27,20)(16,31,33,40)(18,35,34,30)(19,26,37,24)(28,36,39,32);(9,12)(10,16,17,33)(11,19)(13,26)(14,23)(15,28,29,39)(18,36,34,32)(20,38)(21,37)(22,25)(24,27)(30,40,35,31);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,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, 281482);