#193007

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1087 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)(9,10,14,19,15,20,13,18)(11,21,23,16,24,30,28,31)(12,17,25,32,27,33,26,34)(22,35,37,29,38,39,36,40);(2,6)(3,4)(7,8)(9,11,14,23,15,24,13,28)(10,16,19,30,20,31,18,21)(12,22,25,37,27,38,26,36)(17,29,32,39,33,40,34,35);(2,7)(3,4)(6,8)(9,12)(10,17)(11,22)(13,26)(14,25)(15,27)(16,29)(18,34)(19,32)(20,33)(21,35)(23,37)(24,38)(28,36)(30,39)(31,40);(1,3,5,4)(2,7,8,6)(9,13,15,14)(10,18,20,19)(11,23,24,28)(12,25,27,26)(16,30,31,21)(17,32,33,34)(22,36,38,37)(29,35,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100203);