#147552

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#394
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10964
prime radical	C₂×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

991 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=di,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, 54748);