#197548

order	512
nilpotency class	3
isoclinism	isoclinic class of #197339
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,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, 104744);