#479752

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

786 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

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, 386948);