#328367

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 700 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)(27,35)(28,31)(29,33)(30,39)(32,37)(34,38)(36,40);(2,5)(6,8)(9,11,13,21)(10,16,18,12)(14,22,15,17)(19,24,20,23)(25,27,29,28)(26,31,33,35)(30,37,38,36)(32,39,40,34);(3,7)(6,8)(9,12)(10,17)(11,20)(13,16)(14,24)(15,23)(18,22)(19,21)(25,28,29,27)(26,32,33,40)(30,36,38,37)(31,34,35,39);(1,3,4,7)(2,6,5,8)(11,21)(12,16)(17,22)(23,24)(27,28)(31,35)(32,40)(36,37)

Polycyclic presentation:

2^9|b^2=h,c^2=i,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, 235563);