#6443033

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437793
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×#3068	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2538 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)(5,6,9,16,10,17,11,15)(7,18,21,29,22,30,20,12)(8,14,24,32,25,13,26,33)(19,34,23,27,36,31,35,28);(5,7)(6,12)(8,23)(9,20)(10,22)(11,21)(13,31)(14,34)(15,18)(16,30)(17,29)(19,24)(25,35)(26,36)(27,33)(28,32);(1,2,3,4)(5,8)(6,13)(7,19)(9,24)(10,25)(11,26)(12,27)(14,17)(15,32)(16,33)(18,31)(20,35)(21,23)(22,36)(28,29)(30,34);(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,28)(13,17)(15,33)(16,32)(18,34)(20,35)(21,23)(22,36)(27,29)(30,31);(5,9,10,11)(6,15,17,16)(7,20,22,21)(8,24,25,26)(12,18,29,30)(13,32,14,33)(19,35,36,23)(27,31,28,34)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6350229);