#201702

order	512
nilpotency class	3
isoclinism	isoclinic class of #201701
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1885 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 108898);