#426989

order	512
nilpotency class	3
isoclinism	isoclinic class of #418508
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

746 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=f,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,d^b=de,e^b=ei,f^b=fi,h^b=hi,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, 334185);