#147464

order	512
nilpotency class	4
isoclinism	isoclinic class of #12077
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#442
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 54660);