#185848

order	512
nilpotency class	3
isoclinism	isoclinic class of #185517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2158 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)(3,11,13,8)(4,9)(5,7)(6,10)(12,16,14,15)(17,18,22,26,23,27,21,25)(19,28,29,20,24,31,30,32);(2,7)(4,5)(8,15)(9,10)(11,16)(12,14)(17,19,23,24)(18,20,27,32)(21,29,22,30)(25,31,26,28);(1,3,6,13)(2,8,10,11)(4,14,5,12)(7,15,9,16)(17,20)(18,24)(19,27)(21,28)(22,31)(23,32)(25,29)(26,30);(1,4,6,5)(2,9,10,7)(3,12,13,14)(8,15,11,16)(17,21,23,22)(18,25,27,26)(19,29,24,30)(20,31,32,28)

Polycyclic presentation:

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

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, 93044);