#359836

order	512
nilpotency class	3
isoclinism	isoclinic class of #196122
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2172 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,24,27,17)(13,18)(14,20)(15,19)(21,37,39,30)(22,31)(23,32)(25,40,38,33)(26,36,29,34)(28,35);(2,5)(6,8)(9,11)(10,16)(12,25)(13,22)(14,28)(15,23)(17,33)(18,31)(19,32)(20,35)(21,26)(24,40)(27,38)(29,39)(30,34)(36,37);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,34)(19,24)(20,36)(22,38)(23,39)(25,28)(31,40)(32,37)(33,35);(2,5)(6,8)(9,13,15,14)(10,18,19,20)(11,22,23,28)(12,26,27,29)(16,31,32,35)(17,34,24,36)(21,38,39,25)(30,40,37,33)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,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, 267032);