#347046

order	512
nilpotency class	3
isoclinism	isoclinic class of #347042
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2689 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 2689 conjugacy classes of subgroups — too many to draw. Only its 190 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=h,b^a=be,c^a=cf,d^a=dg,f^b=fh,g^b=gi,e^c=eh,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254242);