#102535

order	512
nilpotency class	3
isoclinism	isoclinic class of #102505
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

390 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,10)(4,15)(5,8)(6,9)(7,12)(11,14)(13,16)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26)(33,34,37,40)(35,41,43,48)(36,44,46,39)(38,45,47,42);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,11,15,10)(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(1,2)(3,11)(4,8)(5,15)(6,9)(7,12)(10,14)(13,16)(20,30)(22,31)(23,28)(26,29)(33,35,37,43)(34,38,40,47)(36,45,46,42)(39,48,44,41)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9731);