#104356

order	512
nilpotency class	3
isoclinism	isoclinic class of #104309
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

435 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(2,7)(4,5)(8,15)(9,10)(11,16)(12,14)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28)(33,34,38,41)(35,39,37,45)(36,43,44,47)(40,46,48,42);(1,2,6,9)(3,11,13,15)(4,10,5,7)(8,14,16,12)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31)(33,35,38,37)(34,39,41,45)(36,42,44,46)(40,43,48,47);(1,3,6,13)(2,8,9,16)(4,12,5,14)(7,11,10,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31)(33,36,38,44)(34,40,41,48)(35,42,37,46)(39,47,45,43)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgi,c^b=cf,d^b=dg,e^b=eghi,f^b=fg,d^c=dh,e^c=ei,f^c=fg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11552);