#121585

order	512
nilpotency class	3
isoclinism	isoclinic class of #10254
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

566 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:

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

Polycyclic presentation:

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

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, 28781);