#7614098

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2563 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=h,e^2=f,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,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, 7521294);