#442196

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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, 349392);