#149442

order	512
nilpotency class	4
isoclinism	isoclinic class of #129412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

840 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,g^b=ghi,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56638);