#330949

order	512
nilpotency class	3
isoclinism	isoclinic class of #330923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

589 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,c^b=cg,e^b=eh,g^b=gi,d^c=de,f^c=fh,g^c=ghi,g^d=gh

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, 238145);