#97011

order	512
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#379
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

604 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg

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, 4207);