#330932

order	512
nilpotency class	3
isoclinism	isoclinic class of #330923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

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

normal subgroups only

This group has 620 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)(3,6,8,7)(10,15)(11,19)(13,25)(16,29)(17,27)(20,31)(21,34)(22,26)(23,36)(30,39)(32,40)(35,37);(1,3,4,8)(2,6,5,7)(9,10,13,18,14,15,25,28)(11,20,19,29,22,16,26,31)(12,17,23,33,24,27,36,38)(21,32,34,39,35,30,37,40);(2,5)(3,7)(6,8)(9,11,14,22)(10,16,15,20)(12,21,24,35)(13,26,25,19)(17,32,27,30)(18,29,28,31)(23,37,36,34)(33,40,38,39);(9,12)(10,17)(11,21)(13,23)(14,24)(15,27)(16,30)(18,33)(19,34)(20,32)(22,35)(25,36)(26,37)(28,38)(29,39)(31,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^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, 238128);