#202082

order	512
nilpotency class	3
isoclinism	isoclinic class of #201928
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109278);