#6443314

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4972 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)(7,17)(8,13)(9,14)(10,15)(11,16)(12,20)(18,29)(19,28)(21,27)(22,35)(23,30)(24,32)(25,31)(26,33)(34,36);(5,7,9,20,11,21,10,19)(6,12,15,17,16,28,14,27)(8,22,23,18,24,33,25,34)(13,29,31,35,32,36,30,26);(1,2,3,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,30)(15,31)(16,32)(17,29)(19,22)(20,33)(21,34)(27,36)(28,35);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,30)(15,31)(16,32)(17,29)(19,22)(20,33)(21,34)(27,36)(28,35);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,24,25)(12,17,28,27)(13,30,32,31)(18,22,34,33)(26,29,35,36)

Polycyclic presentation:

2^9|b^2=f,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,e^b=ei,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, 6350510);