#148531

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

682 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)(9,10,15,22,16,24,34,23)(11,25,29,36,31,18,30,39)(12,32,27,37,28,40,26,17)(13,21,14,19,33,38,35,20);(5,6)(7,8)(9,11)(10,17)(12,33)(13,28)(14,26)(15,29)(16,31)(18,38)(19,36)(20,39)(21,25)(22,32)(23,40)(24,37)(27,35)(30,34);(2,4)(6,8)(9,12,13,31,34,27,35,30,16,28,33,11,15,26,14,29)(10,18,19,32,22,36,38,17,24,25,20,40,23,39,21,37)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=hi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,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, 55727);