#103823

order	512
nilpotency class	3
isoclinism	isoclinic class of #103781
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

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

normal subgroups only

This group has 410 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)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18)(19,26)(20,30)(21,24)(22,25)(23,27)(28,31)(29,32)(33,34,37,40)(35,41,43,48)(36,44,46,39)(38,45,47,42);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(17,19,22,28)(18,23,25,32)(20,29,21,27)(24,31,30,26);(6,15)(9,16)(19,27)(20,21)(23,31)(24,30)(26,32)(28,29)(33,35,37,43)(34,38,40,47)(36,45,46,42)(39,48,44,41)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11019);