#196124

order	512
nilpotency class	3
isoclinism	isoclinic class of #196122
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1220 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103320);