#6443316

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5036 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)(7,18)(8,14)(9,15)(10,17)(11,16)(12,21)(13,25)(19,34)(20,29)(22,30)(23,27)(24,33)(26,32)(28,35)(31,36);(5,7,9,21,10,22,11,20)(6,12,16,18,17,29,15,30)(8,23,24,19,25,35,26,36)(13,31,33,28,14,34,32,27);(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)(20,23)(21,35)(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)(20,23)(21,35)(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)(19,23,36,35)(27,31,28,34)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,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, 6350512);