#187167

order	512
nilpotency class	3
isoclinism	isoclinic class of #187159
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	#62752	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2092 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)(3,10,7,4)(5,13)(6,9)(8,14)(11,12,15,16)(18,23,25,28)(24,31,32,30);(1,3,6,7)(2,4,9,10)(5,11,14,15)(8,12,13,16)(17,18)(19,24)(20,23)(21,28)(22,25)(26,30)(27,32)(29,31);(1,4)(2,7)(3,9)(5,12)(6,10)(8,15)(11,13)(14,16)(17,19)(18,24)(20,26)(21,29)(22,27)(23,30)(25,32)(28,31);(1,5,6,14)(2,8,9,13)(3,11,7,15)(4,12,10,16)(17,20,22,21)(18,23,25,28)(19,26,27,29)(24,30,32,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,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, 94363);