#205672

order	512
nilpotency class	3
isoclinism	isoclinic class of #205546
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1216 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)(3,4)(5,8)(6,7)(9,10,14,19,15,20,13,18)(11,21,24,16,25,31,23,32)(12,26,28,34,29,33,27,17)(22,36,38,35,39,30,37,40);(1,3,4,2,6,8,5,7)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,30)(18,31)(19,21)(20,32)(26,35)(27,38)(28,37)(29,39)(33,40)(34,36);(9,12,14,28,15,29,13,27)(10,17,18,33,20,34,19,26)(11,22,23,37,25,39,24,38)(16,30,31,40,32,36,21,35);(1,4,6,5)(2,3,7,8)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112868);