#196101

order	512
nilpotency class	3
isoclinism	isoclinic class of #196101
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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1201 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 103297);