#6442615

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437478
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4517 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)(5,6,12,18)(7,19,21,31)(8,14,25,34)(9,15,27,36)(10,16,11,17)(13,28,32,20)(22,35,26,29)(23,30,24,33);(5,7)(6,13)(8,22)(9,24)(10,20)(11,28)(12,21)(14,15)(16,31)(17,19)(18,32)(23,27)(25,26)(29,33)(30,35)(34,36);(1,2,3,4)(5,8)(6,14)(7,9)(10,24)(11,23)(12,25)(13,29)(15,19)(16,33)(17,30)(18,34)(20,26)(21,27)(22,28)(31,36)(32,35);(5,9)(6,15)(7,8)(10,22)(11,26)(12,27)(13,30)(14,19)(16,35)(17,29)(18,36)(20,23)(21,25)(24,28)(31,34)(32,33);(5,10,12,11)(6,16,18,17)(7,20,21,28)(8,23,25,24)(9,22,27,26)(13,31,32,19)(14,30,34,33)(15,35,36,29)

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6349811);