#7614097

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563327
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

2534 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:

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

Polycyclic presentation:

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

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, 7521293);