#104757

order	512
nilpotency class	3
isoclinism	isoclinic class of #104753
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator, Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

799 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 799 conjugacy classes of subgroups — too many to draw. Only its 76 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)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,27,29,32)(22,24)(23,28,25,31)(26,30);(9,11)(10,13)(12,15)(14,16)(17,19,21,18)(20,23)(22,26,30,24)(25,29)(27,28)(31,32);(2,5)(3,8)(6,7)(13,16)(17,20)(18,23)(19,25)(21,29)(22,31)(24,32)(26,27)(28,30)

Polycyclic presentation:

2^9|a^2=dg,b^2=d,e^2=gh,b^a=bd,c^a=ce,e^a=eh,f^a=fgi,c^b=cf,e^b=ei,f^b=fg,d^c=dg,e^c=egh

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, 11953);