#149483

order	512
nilpotency class	4
isoclinism	isoclinic class of #129617
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×D₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 791 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303570
maximal normal subgroup of	#72303570
maximal quotient of	#72303620

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56679);