#479861

order	512
nilpotency class	3
isoclinism	isoclinic class of #478213
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

574 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,5,7,8)(9,10,15,21,16,22,14,20)(11,19,13,32,25,34,24,17)(12,26,29,38,31,37,30,18)(23,33,27,39,28,36,40,35);(1,3,4,7)(2,5,6,8)(9,11,14,13,16,25,15,24)(10,17,21,34,22,32,20,19)(12,27,30,28,31,40,29,23)(18,35,26,36,38,39,37,33);(3,7)(5,8)(9,12,14,30,16,31,15,29)(10,18,21,26,22,38,20,37)(11,23,13,27,25,28,24,40)(17,33,34,35,32,36,19,39);(9,13,14,25,16,24,15,11)(10,19,20,32,22,34,21,17)(12,28,29,27,31,23,30,40)(18,36,26,39,38,33,37,35)

Polycyclic presentation:

2^9|a^2=efg,b^2=egh,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,g^d=gi,h^d=hi

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, 387057);