#193799

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1293 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100995);