#479812

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

597 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387008);