#149469

order	512
nilpotency class	4
isoclinism	isoclinic class of #129565
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

684 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56665);