#205546

order	512
nilpotency class	3
isoclinism	isoclinic class of #205546
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	#60673	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,e^2=hi,f^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, 112742);