#205647

order	512
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

972 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)(9,10,14,20,15,19,13,18)(11,16,22,30,23,32,28,31)(12,24,26,34,27,33,25,17)(21,35,36,29,37,39,38,40);(1,2,3,5,4,6,7,8)(9,11,14,28,15,23,13,22)(10,16,18,30,19,32,20,31)(12,21,26,38,27,37,25,36)(17,29,33,39,34,40,24,35);(9,12,13,25,15,27,14,26)(10,17,20,24,19,34,18,33)(11,21,22,36,23,37,28,38)(16,29,31,35,32,40,30,39);(9,13,15,14)(10,18,19,20)(11,22,23,28)(12,25,27,26)(16,30,32,31)(17,33,34,24)(21,36,37,38)(29,39,40,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112843);