#329119

order	512
nilpotency class	3
isoclinism	isoclinic class of #327117
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	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

733 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 733 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236315);