#147669

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,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,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54865);