#7563345

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563341
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2642 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=g,e^2=fi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470541);