#6431946

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×#3586	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4511 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:

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

Polycyclic presentation:

2^9|b^2=i,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, 6339142);