#99292

order	512
nilpotency class	3
isoclinism	isoclinic class of #8803
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#383
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

232 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dei,c^2=d,d^2=gh,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fh,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, 6488);