#117581

order	512
nilpotency class	3
isoclinism	isoclinic class of #10549
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

747 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24777);