#453894

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

611 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=efg,d^2=ei,f^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,d^b=de,f^b=fi,h^b=hi,f^c=fi,g^c=gi

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, 361090);