#369411

order	512
nilpotency class	3
isoclinism	isoclinic class of #196143
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1414 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)(4,6)(5,8)(9,10)(11,15)(12,16,24,32)(13,18)(14,26)(17,23)(19,33,35,27)(20,36)(21,28)(22,30,38,40)(25,39,34,31)(29,37);(1,3,5,4)(2,6,8,7)(9,11,13,21)(10,15,18,28)(12,22)(14,20,23,37)(16,30)(17,29,26,36)(19,34)(24,38)(25,35)(27,39)(31,33)(32,40);(1,3,5,4)(2,7,8,6)(9,12,23,34,13,24,14,25)(10,16,17,31,18,32,26,39)(11,19,20,22,21,35,37,38)(15,27,36,40,28,33,29,30);(1,4,5,3)(2,7,8,6)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=efg,d^2=e,g^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, 276607);