#326490

order	512
nilpotency class	3
isoclinism	isoclinic class of #326487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

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

560 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233686);