#6467864

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
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
prime radical	#60700	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1826 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)(11,21)(12,17)(13,18)(14,20)(15,19)(16,23)(22,24);(9,11,14,23,15,17,13,22)(10,16,18,21,19,24,20,12);(2,5)(6,8)(9,12,13,24,15,21,14,16)(10,11,18,23,19,17,20,22);(9,12,14,16,15,21,13,24)(10,17,20,23,19,11,18,22);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,19,20)(11,22,17,23)(12,16,21,24)

Polycyclic presentation:

2^9|b^2=f,c^2=hi,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,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375060);