#192588

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1042 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)(3,6)(4,7)(5,8)(9,10,14,18,15,19,13,29)(11,20,22,16,24,31,23,32)(12,25,26,17,28,33,27,34)(21,36,38,35,39,30,37,40);(2,6)(3,4)(7,8)(9,11,14,22,15,24,13,23)(10,16,18,31,19,32,29,20)(12,21,26,38,28,39,27,37)(17,30,33,40,34,36,25,35);(2,7)(3,4)(6,8)(9,12)(10,17)(11,21)(13,27)(14,26)(15,28)(16,30)(18,33)(19,34)(20,35)(22,38)(23,37)(24,39)(25,29)(31,40)(32,36);(1,3,5,4)(2,6,8,7)(9,13,15,14)(10,18,19,29)(11,22,24,23)(12,26,28,27)(16,20,32,31)(17,25,34,33)(21,37,39,38)(30,40,36,35)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fh

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, 99784);