#330310

order	512
nilpotency class	3
isoclinism	isoclinic class of #329647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 615 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)(9,10,15,21,14,12,13,17)(11,16,22,24,19,18,20,23)(25,26,30,32,29,33,28,34)(27,31,37,39,36,40,35,38);(2,5)(6,8)(9,11)(10,16)(12,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,40)(34,38);(3,7)(6,8)(9,12,14,10)(11,18,19,16)(13,17,15,21)(20,23,22,24)(26,32)(27,35)(28,30)(33,34)(36,37)(38,39);(1,3,4,7)(2,6,5,8)(11,19)(16,18)(20,22)(23,24)(26,33)(27,36)(32,34)(35,37)

Polycyclic presentation:

2^9|a^2=fh,c^2=hi,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, 237506);