#7518819

order	512
nilpotency class	3
isoclinism	isoclinic class of #7513980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(2,3)(9,10,14,11)(12,19,22,20)(13,15,23,17)(16,24,18,21);(1,2,4,3)(5,6)(7,8)(9,11,14,10)(12,20,22,19)(13,17,23,15)(16,21,18,24);(10,15)(11,17)(13,23)(16,19)(18,20)(21,24);(6,8)(9,12,14,22)(10,16,11,18)(13,24,23,21)(15,19,17,20);(1,3,4,2)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315301, #72315344, #72315350
maximal normal subgroup of	#72315301, #72315344, #72315350
maximal quotient of	#72315713

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426015);