#99287

order	512
nilpotency class	3
isoclinism	isoclinic class of #99283
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	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

242 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483);