#359814

order	512
nilpotency class	3
isoclinism	isoclinic class of #196338
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2168 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)(9,10,14,20,15,19,13,18)(11,16,24,32,23,31,22,30)(12,25,26,35,27,17,28,34)(21,29,36,39,37,33,38,40);(2,5)(6,8)(9,11)(10,16)(12,21)(13,22)(14,24)(15,23)(17,33)(18,30)(19,31)(20,32)(25,29)(26,36)(27,37)(28,38)(34,40)(35,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,29)(18,34)(19,25)(20,35)(22,36)(23,37)(24,38)(30,39)(31,33)(32,40);(2,5)(6,8)(9,13,15,14)(10,18,19,20)(11,22,23,24)(12,26,27,28)(16,30,31,32)(17,34,25,35)(21,36,37,38)(29,39,33,40)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,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, 267010);