#479672

order	512
nilpotency class	3
isoclinism	isoclinic class of #478457
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1085 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1085 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386868);