#103209

order	512
nilpotency class	3
isoclinism	isoclinic class of #103197
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

1001 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 1001 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,d^c=dh,e^c=eg

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10405);