#201907

order	512
nilpotency class	3
isoclinism	isoclinic class of #201717
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#37452	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1249 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109103);