#201901

order	512
nilpotency class	3
isoclinism	isoclinic class of #201701
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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1312 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,g^2=hi,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, 109097);