#288043

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28485	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

866 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efhi,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,g^c=gi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195239);