#147617

order	512
nilpotency class	4
isoclinism	isoclinic class of #147614
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10966
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

869 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gh,f^c=fi,g^c=gi,g^d=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54813);