#6443037

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
prime radical	#60426	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2511 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)(5,6,11,16,12,7,10,15)(8,13,26,19,21,17,22,29)(9,14,27,23,25,18,24,30)(20,28,35,32,33,31,34,36);(5,7,12,6)(8,19,21,29)(9,23,25,30)(10,15,11,16)(13,22,17,26)(14,24,18,27)(20,31,33,28)(32,34,36,35);(1,2,3,4)(5,8)(6,13)(7,17)(9,20)(10,22)(11,26)(12,21)(14,28)(15,29)(16,19)(18,31)(23,32)(24,34)(25,33)(27,35)(30,36);(5,9)(6,14)(7,18)(8,20)(10,24)(11,27)(12,25)(13,28)(15,30)(16,23)(17,31)(19,32)(21,33)(22,34)(26,35)(29,36);(8,21)(13,17)(19,29)(20,33)(22,26)(28,31)(32,36)(34,35)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=g,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, 6350233);