#6467263

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442584
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2009 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320791
maximal normal subgroup of	#72320791
maximal quotient of	#72321621

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374459);