#6431883

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	#62859	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4605 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)(7,11)(8,13)(9,12)(10,15)(14,22)(16,24)(17,23)(18,26)(19,28)(20,27)(21,25)(29,32)(30,35)(31,34)(33,36);(1,2)(3,4)(5,7,10,18)(6,11,15,26)(8,16,19,30)(9,21,22,17)(12,25,14,23)(13,24,28,35)(20,33,32,31)(27,36,29,34);(1,3,4,2)(6,12)(9,22)(11,23)(13,27)(14,15)(17,21)(20,32)(24,34)(25,26)(28,29)(31,33)(35,36);(5,8)(6,13)(7,16)(9,20)(10,19)(11,24)(12,27)(14,29)(15,28)(17,31)(18,30)(21,33)(22,32)(23,34)(25,36)(26,35);(8,19)(13,28)(16,30)(20,32)(24,35)(27,29)(31,33)(34,36)

Polycyclic presentation:

2^9|b^2=i,c^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,g^b=gi,g^c=gi,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, 6339079);