#414581

order	512
nilpotency class	3
isoclinism	isoclinic class of #393929
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	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1932 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1932 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321777);