#196266

order	512
nilpotency class	3
isoclinism	isoclinic class of #196266
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

1287 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239301, #72257630, #72257639
maximal normal subgroup of	#72239301, #72257630, #72257639
maximal quotient of	#72239307

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103462);