#479825

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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

549 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)(3,8)(4,6)(5,7)(9,10,15,21,16,22,14,20)(11,18,12,26,25,34,24,17)(13,19,30,36,31,37,29,35)(23,33,38,32,28,40,27,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,17,22,26)(12,14)(13,28,31,23)(15,24)(16,25)(18,21,34,20)(19,32)(27,30,38,29)(33,36)(35,40)(37,39);(1,3,5,4)(2,6,7,8)(9,12,14,11,16,24,15,25)(10,18,21,26,22,34,20,17)(13,23,30,38,31,28,29,27)(19,32,35,33,37,39,36,40);(1,4,5,3)(2,6,7,8)(9,13,15,29,16,31,14,30)(10,19,20,36,22,37,21,35)(11,23,12,27,25,28,24,38)(17,32,34,40,26,39,18,33)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,e^c=ei,e^d=ei,g^d=gi,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, 387021);