#413817

order	512
nilpotency class	3
isoclinism	isoclinic class of #403352
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1477 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 1477 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.

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

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321013);