#205723

order	512
nilpotency class	3
isoclinism	isoclinic class of #205569
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

944 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)(3,7)(6,8)(9,10,14,19,15,18,13,27)(11,16,22,32,21,30,28,31)(12,23,26,33,24,37,25,17)(20,34,36,29,35,39,38,40);(1,2,3,5,4,6,7,8)(9,11,14,28,15,21,13,22)(10,16,27,32,18,30,19,31)(12,20,26,38,24,35,25,36)(17,29,37,39,33,40,23,34);(9,12,13,25,15,24,14,26)(10,17,19,23,18,33,27,37)(11,20,22,36,21,35,28,38)(16,29,31,34,30,40,32,39);(2,6)(5,8)(11,21)(12,24)(16,30)(17,33)(22,28)(23,37)(25,26)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,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,d^c=di

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, 112919);