#149475

order	512
nilpotency class	4
isoclinism	isoclinic class of #129627
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×Q₈	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

799 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 799 conjugacy classes of subgroups — too many to draw. Only its 138 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)(11,21,24,36)(12,18,13,27)(14,19)(15,20)(16,22)(17,26,34,28)(23,30,39,40)(25,32)(29,33)(31,37)(35,38);(1,2,4,6)(3,7,8,5)(9,11)(10,16,20,32)(12,25)(13,22)(14,23)(15,24)(17,30,34,40)(18,21,27,36)(19,35,33,31)(26,38)(28,37)(29,39);(1,3,4,8)(2,5,6,7)(9,12,15,13)(10,17)(11,22,24,25)(14,26,29,28)(16,30)(18,19)(20,34)(21,35)(23,37,39,38)(27,33)(31,36)(32,40)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=f,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=eh,g^b=ghi,d^c=dh,g^c=gi,e^d=ei,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, 56671);