#7563333

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
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

2683 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=g,e^2=f,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,e^c=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, 7470529);