#201511

order	512
nilpotency class	3
isoclinism	isoclinic class of #201424
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	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1343 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=hi,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,f^b=fi,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, 108707);