#6629039

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

4456 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)(6,11)(7,15)(9,21)(13,23)(14,24)(16,29)(17,18)(20,32)(22,27)(28,34)(30,31)(33,35);(5,6,9,11,10,14,21,24)(7,12,17,22,18,26,15,27)(8,13,20,23,19,28,32,34)(16,25,31,33,30,36,29,35);(5,7)(6,12)(8,16)(9,17)(10,18)(11,22)(13,25)(14,26)(15,21)(19,30)(20,31)(23,33)(24,27)(28,36)(29,32)(34,35);(2,4)(5,8)(6,13)(7,16)(9,20)(10,19)(11,23)(12,25)(14,28)(15,29)(17,31)(18,30)(21,32)(22,33)(24,34)(26,36)(27,35);(6,14)(8,19)(11,24)(12,26)(16,30)(20,32)(22,27)(29,31)

Polycyclic presentation:

2^9|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, 6536235);