#7544137

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1964 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282016, #72282375, #72282412
maximal normal subgroup of	#72282016, #72282375, #72282412
maximal quotient of	#72305695, #72305699, #72315840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7451333);