#374209

order	512
nilpotency class	3
isoclinism	isoclinic class of #196266
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1529 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)(11,16)(12,24,27,35)(13,18)(14,19)(15,20)(17,26,33,29)(21,36,25,39)(22,31)(23,32)(28,34)(30,38,40,37);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,25)(13,22)(14,28)(15,23)(17,30)(18,31)(19,34)(20,32)(21,27)(24,39)(26,38)(29,37)(33,40)(35,36);(9,12,14,29,15,27,13,26)(10,17,19,35,20,33,18,24)(11,21,22,37,23,25,28,38)(16,30,31,39,32,40,34,36);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,22,23,28)(12,26,27,29)(16,31,32,34)(17,24,33,35)(21,37,25,38)(30,39,40,36)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=i,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, 281405);