#413195

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2308 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 2308 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,d^2=h,e^2=i,f^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^c=ei,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320391);