#6574233

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

2508 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)(7,13)(8,15)(9,16)(10,14)(11,17)(12,18)(19,28)(20,27)(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)(8,11,19,10)(9,23,25,22)(14,15,17,28)(16,32,34,31)(20,26,35,24)(27,33,36,30);(6,14)(7,19)(8,21)(10,11)(15,28)(16,30)(17,18)(20,23)(22,35)(24,26)(31,32)(33,34);(5,8,12,19)(6,15,18,28)(7,10,21,11)(9,20,25,35)(13,14,29,17)(16,27,34,36)(22,26,23,24)(30,31,33,32);(5,9,12,25)(6,16,18,34)(7,20,21,35)(8,22,19,23)(10,24,11,26)(13,27,29,36)(14,30,17,33)(15,31,28,32)

Polycyclic presentation:

2^9|b^2=fi,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, 6481429);