#350480

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

932 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 168 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,21)(13,17)(14,18)(16,20)(19,24)(22,23)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,39,40,38);(2,5)(6,8)(9,11,14,13)(10,15,18,17)(12,19,22,20)(16,21,24,23)(25,27,30,26)(28,33,36,31)(29,32,37,34)(35,38,40,39);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,20)(14,22)(15,21)(17,23)(18,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(9,12)(10,16)(11,20)(13,19)(14,22)(15,23)(17,21)(18,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=e,d^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,d^b=dhi,f^c=fh,g^c=gi,f^d=fi,g^d=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257676);