#6573985

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

2528 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)(7,13)(8,15)(9,16)(10,14)(11,17)(12,18)(19,27)(20,28)(21,29)(22,31)(23,32)(24,30)(25,34)(26,33)(35,36);(1,2,3,4)(5,7,12,21)(6,13,18,29)(8,10,20,11)(9,22,25,35)(14,28,17,15)(16,31,34,36)(19,26,23,24)(27,33,32,30);(6,14)(7,8)(10,11)(15,28)(16,30)(17,18)(19,22)(20,21)(23,35)(24,26)(31,36)(33,34);(5,8)(6,15)(7,10)(9,23)(11,21)(12,20)(13,14)(16,32)(17,29)(18,28)(19,25)(22,24)(26,35)(27,34)(30,31)(33,36);(5,9)(6,16)(7,19)(8,22)(10,24)(11,26)(12,25)(13,27)(14,30)(15,31)(17,33)(18,34)(20,35)(21,23)(28,36)(29,32)

Polycyclic presentation:

2^9|b^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481181);