#479838

order	512
nilpotency class	3
isoclinism	isoclinic class of #478195
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 792 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=f,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,h^c=hi,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, 387034);