#7519109

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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

2855 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)(11,18)(12,23)(13,20)(14,16)(15,19)(17,21)(22,24);(1,2,4,3)(5,6)(7,8)(10,16)(11,21)(12,22)(13,15)(17,24)(18,23);(9,11)(10,17)(12,13)(14,22)(15,21)(16,23)(18,20)(19,24);(6,8)(9,12,14,21)(10,18,16,24)(11,13,22,15)(17,20,23,19);(1,3,4,2)(9,13,14,15)(10,19,16,20)(11,12,22,21)(17,24,23,18)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263893, #72263918, #72263951
maximal normal subgroup of	#72263893, #72263918, #72263951
maximal quotient of	#72264150, #72264516

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426305);