#189040

order	512
nilpotency class	3
isoclinism	isoclinic class of #188966
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

827 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:

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

Polycyclic presentation:

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

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, 96236);