#6434726

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432953
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3168 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 3168 conjugacy classes of subgroups — too many to draw. Only its 610 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,22)(20,23)(21,24)(25,31)(26,29)(27,30)(28,32);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,15,16,13)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(2,8)(6,14)(7,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32);(18,24)(22,29)(23,30)(28,31)

Polycyclic presentation:

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

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, 6341922);