#205578

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	#60673	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1065 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=hi,f^2=i,g^2=h,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, 112774);