#328364

order	512
nilpotency class	3
isoclinism	isoclinic class of #326925
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	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

698 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235560);