#330936

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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 655 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=h,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, 238132);