#136615

order	512
nilpotency class	4
isoclinism	isoclinic class of #13005
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

415 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:

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

Polycyclic presentation:

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

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, 43811);