#442195

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

623 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 349391);