#187773

order	512
nilpotency class	3
isoclinism	isoclinic class of #187766
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3511	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2345 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=ghi,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, 94969);