#149317

order	512
nilpotency class	4
isoclinism	isoclinic class of #140671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,f^b=fi,g^b=gh,f^c=fi,g^c=gh,g^d=gi,g^e=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, 56513);