#197529

order	512
nilpotency class	3
isoclinism	isoclinic class of #197330
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	#35050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1737 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,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,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, 104725);