#7519166

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2778 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)(11,19)(12,17)(13,15)(14,18)(16,20)(21,23)(22,24);(1,2,4,3)(5,6)(7,8)(9,10)(11,19)(12,17)(13,15)(14,18)(16,20)(21,23)(22,24);(10,15)(12,13)(16,19)(17,18)(20,22)(23,24);(6,8)(9,11,14,21)(10,16,18,24)(12,20,13,22)(15,19,17,23);(1,3,4,2)(9,12,14,13)(10,17,18,15)(11,20,21,22)(16,23,24,19)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315307, #72315342, #72315347
maximal normal subgroup of	#72315307, #72315342, #72315347
maximal quotient of	#72315807

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426362);