#148326

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
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

680 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:

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

Polycyclic presentation:

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

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, 55522);