#6443038

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2530 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350234);