#494781

order	512
nilpotency class	3
isoclinism	isoclinic class of #11247
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#387
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1917
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,13,7,12,28,40,8,11,29,39,5,24,26,42)(3,14,17,44,16,9,30,33,18,34,49,61,25,32,54,59)(4,10,22,38,23,37,53,63,21,36,27,41,20,52,45,56)(15,43,47,64,19,31,51,58,46,35,48,62,50,57,55,60);(1,3,6,17,7,16,28,30,8,18,29,49,5,25,26,54)(2,9,13,33,12,34,40,61,11,32,39,59,24,14,42,44)(4,19,22,51,23,46,53,48,21,50,27,55,20,15,45,47)(10,35,38,62,37,57,63,60,36,43,41,64,52,31,56,58);(1,4,5,20,8,21,7,23)(2,10,12,37,11,36,24,52)(3,15,16,19,18,46,25,50)(6,27,26,53,29,22,28,45)(9,31,14,43,32,57,34,35)(13,41,40,56,39,38,42,63)(17,48,30,55,49,47,54,51)(33,60,44,62,59,58,61,64);(2,11)(9,32)(10,36)(12,24)(13,39)(14,34)(31,57)(33,59)(35,43)(37,52)(38,41)(40,42)(44,61)(56,63)(58,60)(62,64)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,c^b=ch,e^b=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401977);