#120705

order	512
nilpotency class	3
isoclinism	isoclinic class of #117904
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
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

512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=di,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fi,g^b=gh,f^c=fi,g^c=gh

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, 27901);