#6434722

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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3516 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 3516 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) = 32

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,b^a=bf,c^a=cg,d^a=dh,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, 6341918);