#330953

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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 570 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)(9,10,14,19)(11,20,21,15)(12,17,25,32)(13,16,26,18)(22,33,34,28)(23,27,35,29)(24,30,37,31)(36,38,40,39);(2,5)(3,6)(7,8)(9,11,13,23,14,21,26,35)(10,15,18,27,19,20,16,29)(12,22,24,36,25,34,37,40)(17,28,31,38,32,33,30,39);(1,3,4,7)(2,6,5,8)(10,16)(11,21)(13,26)(15,27)(17,30)(18,19)(20,29)(24,37)(28,39)(31,32)(33,38)(36,40);(3,7)(6,8)(9,12,14,25)(10,17,19,32)(11,22,21,34)(13,24,26,37)(15,28,20,33)(16,30,18,31)(23,36,35,40)(27,38,29,39)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=h,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, 238149);