#453819

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

668 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=ef,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,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, 361015);