#6437607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5896 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,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, 6344803);