#6431891

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431723
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,h^b=hi,g^c=gi,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, 6339087);