#6629040

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629034
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	#60371	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4408 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)(5,6,11,15)(7,16,18,32)(8,20,24,29)(9,13,25,19)(10,26,27,14)(12,22,30,23)(17,21,33,36)(28,31,34,35);(5,7,10,19,11,18,27,13)(6,9,14,16,15,25,26,32)(8,17,22,34,24,33,23,28)(12,21,29,31,30,36,20,35);(5,8)(6,12)(7,17)(9,21)(10,22)(11,24)(13,28)(14,29)(15,30)(16,31)(18,33)(19,34)(20,26)(23,27)(25,36)(32,35);(2,4)(5,9)(6,13)(7,14)(8,21)(10,16)(11,25)(12,28)(15,19)(17,29)(18,26)(20,33)(22,31)(23,35)(24,36)(27,32)(30,34);(7,18)(9,25)(13,19)(16,32)(17,33)(21,36)(28,34)(31,35)

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,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, 6536236);