#326524

order	512
nilpotency class	3
isoclinism	isoclinic class of #326524
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2563
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

504 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,f^a=fh,c^b=cg,d^b=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233720);