#116485

order	512
nilpotency class	3
isoclinism	isoclinic class of #116481
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

689 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23681);