#6574232

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
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

2516 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)(7,19)(8,15)(9,16)(10,17)(11,14)(12,18)(13,21)(20,35)(22,29)(23,31)(24,28)(25,33)(26,30)(27,34)(32,36);(1,2,3,4)(5,7,10,21)(6,13,17,19)(8,11,22,12)(9,23,25,36)(14,15,18,29)(16,32,33,31)(20,27,24,26)(28,34,35,30);(6,14)(7,8)(11,12)(15,29)(16,30)(17,18)(20,23)(21,22)(24,36)(26,27)(31,32)(33,34);(5,8,10,22)(6,15,17,29)(7,12,21,11)(9,24,25,20)(13,14,19,18)(16,28,33,35)(23,27,36,26)(30,31,34,32);(5,9,10,25)(6,16,17,33)(7,20,21,24)(8,23,22,36)(11,26,12,27)(13,28,19,35)(14,30,18,34)(15,31,29,32)

Polycyclic presentation:

2^9|b^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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, 6481428);