#120684

order	512
nilpotency class	3
isoclinism	isoclinic class of #10362
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=i,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,f^b=fhi,g^b=ghi,f^c=fhi,g^c=ghi

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, 27880);