#6434718

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432949
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 ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3733 conjugacy classes of subgroups; only the 614 normal subgroups are shown.

normal subgroups only

This group has 3733 conjugacy classes of subgroups — too many to draw. Only its 614 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:

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

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

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, 6341914);