#6573409

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533135
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4420 conjugacy classes of subgroups; only the 586 normal subgroups are shown.

normal subgroups only

This group has 4420 conjugacy classes of subgroups — too many to draw. Only its 586 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) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 16 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=h,e^2=h,b^a=bf,c^a=cg,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339533, #72339539, #72339565
maximal normal subgroup of	#72339533, #72339539, #72339565

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480605);