#186346

order	512
nilpotency class	3
isoclinism	isoclinic class of #186162
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62838	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2049 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:

(1,2,6,9,7,4,5,14)(3,10,12,8,13,16,15,11)(17,18,22,26,23,27,21,25)(19,28,29,20,24,31,30,32);(1,3,7,13)(2,8,4,11)(5,15,6,12)(9,16,14,10)(17,19,23,24)(18,20,27,32)(21,29,22,30)(25,31,26,28);(1,4)(2,7)(3,11)(5,14)(6,9)(8,13)(10,15)(12,16)(17,20)(18,24)(19,27)(21,28)(22,31)(23,32)(25,29)(26,30);(2,4)(8,11)(9,14)(10,16)(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=i,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,g^b=ghi,d^c=di,f^c=fh,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, 93542);