#99281

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#382
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

256 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 56 points:

(1,2,4,6,3,5,7,8)(9,10)(11,19)(12,15)(13,23)(14,18)(16,21)(17,20)(22,24)(25,26)(27,35,39,31)(28,40,42,55)(29,34)(30,33)(32,41,46,50)(36,43)(37,51)(38,48,49,45)(44,52)(47,56)(53,54);(9,11,20,22,14,21,13,12)(10,15,23,16,18,24,17,19)(25,27,36,49,30,39,29,38)(26,31,43,48,33,35,34,45)(28,37,50,56,42,53,41,52)(32,44,55,51,46,47,40,54);(2,5)(6,8)(9,12,13,21,14,22,20,11)(10,16,17,15,18,19,23,24)(25,28)(26,32,33,46)(27,37)(29,41)(30,42)(31,44,35,47)(34,40,43,55)(36,50)(38,52)(39,53)(45,54,48,51)(49,56)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=d,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fgh,f^c=fgh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6477);