#6443039

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	C₂×#3067	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2482 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:

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

Polycyclic presentation:

2^9|a^2=fi,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, 6350235);