#147150

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#386
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10876
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

517 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257881, #72265425, #72265447
maximal normal subgroup of	#72257881, #72265425, #72265447

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54346);