#149306

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#397
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11346
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

467 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,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, 56502);