#326731

order	512
nilpotency class	3
isoclinism	isoclinic class of #326683
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 158 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 158 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)(9,10,13,17)(11,19,22,15)(12,16,14,18)(20,23,21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,40)(36,38)(37,39);(2,5)(6,8)(9,11,13,22)(10,15,17,19)(12,21,14,20)(16,23,18,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,40)(33,38)(34,39);(1,3)(2,6)(4,7)(5,8)(10,16)(11,20)(12,14)(17,18)(21,22)(23,24)(25,28,30,29)(26,32,34,33)(27,35,37,36)(31,38,39,40);(3,7)(6,8)(9,12,13,14)(10,16,17,18)(11,21,22,20)(15,23,19,24)(25,29,30,28)(26,33,34,32)(27,36,37,35)(31,38,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233927);