#7563300

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519075
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

2626 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)(8,14)(9,15)(10,25)(11,17)(12,18)(13,22)(16,26)(20,29)(21,31)(23,33)(24,28)(27,32)(30,35)(34,36);(2,3)(5,7,11,22,12,9,8,20)(6,13,14,19,18,29,17,15)(10,24,26,35,27,21,23,36)(16,31,25,30,33,28,32,34);(1,2,4,3)(5,8,12,11)(6,14,18,17)(7,20,9,22)(10,23,27,26)(13,19,29,15)(16,25,33,32)(21,35,24,36)(28,34,31,30);(1,3,4,2)(5,9)(6,15)(7,12)(8,22)(10,24)(11,20)(13,14)(16,30)(17,29)(18,19)(21,27)(23,36)(25,28)(26,35)(31,32)(33,34);(5,10,11,26,12,27,8,23)(6,16,14,25,18,33,17,32)(7,21,22,36,9,24,20,35)(13,28,19,34,29,31,15,30)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=g,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,e^b=ei,h^b=hi,h^d=hi

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, 7470496);