#406404

order	512
nilpotency class	3
isoclinism	isoclinic class of #398436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1313 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)(3,6)(4,5)(7,8)(9,10,15,19)(11,20,24,37)(12,17,26,27)(13,25,28,39)(14,29,30,18)(16,31,33,23)(21,22,38,35)(32,36,34,40);(2,5)(6,8)(9,11,14,23,15,24,30,31)(10,16,18,20,19,33,29,37)(12,21,25,36,26,38,39,40)(13,22,27,32,28,35,17,34);(9,12)(10,17)(11,21,24,38)(13,18)(14,25)(15,26)(16,32,33,34)(19,27)(20,35,37,22)(23,36,31,40)(28,29)(30,39);(1,3,4,7)(2,6,5,8)(9,13,14,27,15,28,30,17)(10,12,18,25,19,26,29,39)(11,22,23,32,24,35,31,34)(16,21,20,36,33,38,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313600);