#6431945

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3585	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4597 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:

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

Polycyclic presentation:

2^9|c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,f^b=fi,g^b=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, 6339141);