#104378

order	512
nilpotency class	3
isoclinism	isoclinic class of #104373
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	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

474 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=g,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=egi,f^b=fh,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11574);