#415287

order	512
nilpotency class	3
isoclinism	isoclinic class of #397258
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

800 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 234 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=ef,c^2=e,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322483);