#148618

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

658 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^b=gi,e^c=eh,h^c=hi

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, 55814);