#459030

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

771 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366226);