#6432121

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(5,6,10,18,12,17,11,16)(7,14,20,13,8,22,21,28)(9,15,25,32,24,30,26,31)(19,29,33,27,23,35,34,36);(1,2)(3,4)(5,7,10,20,12,8,11,21)(6,13,16,14,17,28,18,22)(9,19,25,33,24,23,26,34)(15,27,31,29,30,36,32,35);(2,4)(5,8,12,7)(6,14,17,22)(9,23,24,19)(10,20,11,21)(13,18,28,16)(15,29,30,35)(25,33,26,34)(27,32,36,31);(5,9)(6,15)(7,19)(8,23)(10,25)(11,26)(12,24)(13,27)(14,29)(16,31)(17,30)(18,32)(20,33)(21,34)(22,35)(28,36);(9,24)(15,30)(19,23)(25,26)(27,36)(29,35)(31,32)(33,34)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,f^c=fi,h^c=hi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339317);