#412912

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

780 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)(3,5)(4,6)(7,8)(9,10,13,17,14,19,27,18)(11,15,22,29,23,20,33,30)(12,24,25,16,26,31,38,32)(21,35,36,28,37,39,40,34);(1,3,4,7)(2,5,6,8)(9,10)(11,20)(12,16)(13,18)(14,19)(15,23)(17,27)(21,34)(22,29)(24,25)(26,32)(28,37)(30,33)(31,38)(35,40)(36,39);(2,6)(5,8)(9,11,13,22,14,23,27,33)(10,15,17,29,19,20,18,30)(12,21,25,36,26,37,38,40)(16,28,31,39,32,34,24,35);(9,12,13,25,14,26,27,38)(10,16,18,24,19,32,17,31)(11,21,22,36,23,37,33,40)(15,28,30,35,20,34,29,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320108);