#116669

order	512
nilpotency class	3
isoclinism	isoclinic class of #116666
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

576 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23865);