#101710

order	512
nilpotency class	3
isoclinism	isoclinic class of #101687
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

362 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gh,e^2=hi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8906);