#193109

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#121	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

868 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)(9,10,14,19,15,18,13,29)(11,20,22,16,24,32,23,31)(12,25,27,17,28,34,26,33)(21,36,37,35,38,30,39,40);(3,7)(5,8)(9,11,14,22,15,24,13,23)(10,16,19,32,18,31,29,20)(12,21,27,37,28,38,26,39)(17,30,34,40,33,36,25,35);(1,2)(3,8)(4,6)(5,7)(9,12)(10,17)(11,21)(13,26)(14,27)(15,28)(16,30)(18,33)(19,34)(20,35)(22,37)(23,39)(24,38)(25,29)(31,36)(32,40);(10,18)(16,31)(17,33)(19,29)(20,32)(25,34)(30,36)(35,40)

Polycyclic presentation:

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

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, 100305);