#103856

order	512
nilpotency class	3
isoclinism	isoclinic class of #103845
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	#2867	C₂²
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

520 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=g,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,d^c=dh,e^c=ei,f^c=fgh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11052);