#330301

order	512
nilpotency class	3
isoclinism	isoclinic class of #329647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#30258	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

720 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 138 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)(9,10,12,16,13,17,20,15)(11,14,19,23,18,22,24,21)(25,26)(27,30)(28,32)(29,31)(33,37)(34,39)(35,38)(36,40);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24)(25,27,29,35)(26,30,31,38)(28,33,36,34)(32,37,40,39);(3,7)(6,8)(10,15)(12,20)(14,21)(16,17)(19,24)(22,23)(27,33)(28,36)(30,37)(32,40)(34,35)(38,39);(1,3,4,7)(2,6,5,8)(11,18)(14,22)(19,24)(21,23)(26,31)(30,38)(32,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237497);