#106404

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

984 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13600);