#328314

order	512
nilpotency class	3
isoclinism	isoclinic class of #326835
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)(9,10,15,19)(11,12,20,23)(13,17,14,18)(16,21,24,22)(25,26)(27,31)(28,33)(29,37)(30,34)(32,35)(36,40)(38,39);(2,5)(6,8)(9,11,15,20)(10,12,19,23)(13,16,14,24)(17,21,18,22)(25,27,30,36)(26,31,34,40)(28,35,29,38)(32,37,39,33);(3,7)(6,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,20)(19,24)(25,27)(26,32)(28,35)(29,38)(30,36)(31,37)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,17,19,18)(11,16,20,24)(12,21,23,22)(25,28,30,29)(26,33,34,37)(27,35,36,38)(31,32,40,39)

Polycyclic presentation:

2^9|a^2=hi,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,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, 235510);