#148459

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

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	#11147
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

637 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,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,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, 55655);