#6499867

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499867
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

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 ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

11818 conjugacy classes of subgroups; only the 598 normal subgroups are shown.

normal subgroups only

This group has 11818 conjugacy classes of subgroups — too many to draw. Only its 598 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) = 16

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72227565, #72227786, #72227864, …
maximal subgroup of	#72209716, #72209717, #72209835, …
maximal normal subgroup of	#72209716, #72209717, #72209835, …
maximal quotient of	#72209710, #72210404, #72210417, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407063);