#424646

order	512
nilpotency class	3
isoclinism	isoclinic class of #417454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1553 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331842);