#121620

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

528 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28816);