#104628

order	512
nilpotency class	3
isoclinism	isoclinic class of #104565
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

403 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=g,e^2=i,f^2=h,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=fh,d^c=dh,e^c=ei,f^c=fh

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, 11824);