#149316

order	512
nilpotency class	4
isoclinism	isoclinic class of #140671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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

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

normal subgroups only

This group has 624 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)(11,19)(12,24,26,16)(13,17)(14,18)(15,21,30,39)(20,28)(22,35)(23,36)(25,33,40,31)(27,34)(29,38,32,37);(1,3,5,4)(2,6,7,8)(9,11,14,23)(10,15,18,30)(12,21,26,39)(13,20,27,22)(16,19,24,36)(17,32,34,29)(25,37,40,38)(28,33,35,31);(1,4,5,3)(2,6,7,8)(10,16)(11,20)(13,27)(15,28)(17,33)(18,24)(19,32)(21,37)(22,23)(25,40)(29,36)(30,35)(31,34)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=dei,c^2=de,f^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, 56512);