#352431

order	512
nilpotency class	3
isoclinism	isoclinic class of #347698
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

662 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^c=dh,g^d=gi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259627);