#190184

order	512
nilpotency class	3
isoclinism	isoclinic class of #190116
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1044 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:

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

Polycyclic presentation:

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

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, 97380);