#396046

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2034 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^c=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303242);