#6573994

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
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

2524 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)(5,6,12,18)(7,13,21,29)(8,22,19,15)(9,16,26,34)(10,14,11,17)(20,28,35,32)(23,36,24,31)(25,30,27,33);(1,2,3,4)(5,7)(6,13)(8,10)(9,24)(11,19)(12,21)(14,22)(15,17)(16,31)(18,29)(20,25)(23,26)(27,35)(28,30)(32,33)(34,36);(6,14)(7,19)(8,21)(10,11)(15,22)(16,30)(17,18)(20,24)(23,35)(25,27)(31,36)(33,34);(5,8,12,19)(6,15,18,22)(7,10,21,11)(9,20,26,35)(13,17,29,14)(16,32,34,28)(23,27,24,25)(30,31,33,36);(5,9)(6,16)(7,20)(8,23)(10,25)(11,27)(12,26)(13,28)(14,30)(15,31)(17,33)(18,34)(19,24)(21,35)(22,36)(29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481190);