#6629033

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629032
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	#60359	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4487 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)(5,6,11,16)(7,17,19,28)(8,22,24,30)(9,15,10,14)(12,21,27,20)(13,25,29,23)(18,32,33,26)(31,35,36,34);(5,7,9,20,11,19,10,21)(6,12,14,17,16,27,15,28)(8,18,23,34,24,33,25,35)(13,26,22,31,29,32,30,36);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,22)(15,30)(16,29)(17,31)(19,33)(20,34)(21,35)(27,32)(28,36);(2,4);(7,19)(12,27)(17,28)(18,33)(20,21)(26,32)(31,36)(34,35)

Polycyclic presentation:

2^9|a^2=i,b^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,e^b=ei,g^c=gi

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, 6536229);