#147746

order	512
nilpotency class	4
isoclinism	isoclinic class of #147745
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10980
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1028 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)(3,15,20,43)(4,21)(5,27,30,11)(6,31,32,12)(7,13)(8,14)(9,17)(10,24,42,56)(16,39,50,62)(18,37)(19,36,33,60)(22,54,52,35)(23,46,57,41)(25,40)(26,53)(28,38)(29,44)(34,58)(45,51)(47,55)(48,61)(49,64)(59,63);(1,3,8,20)(2,9,14,38)(4,22,26,52)(5,28,30,17)(6,18,32,51)(7,33,29,19)(10,39,42,62)(11,43,27,15)(12,36,31,60)(13,45,44,37)(16,34,50,25)(21,47,53,64)(23,49,57,55)(24,48,56,59)(35,46,54,41)(40,61,58,63);(1,4)(2,10)(3,16)(5,23)(6,24)(7,34)(8,26)(9,35)(11,40)(12,21)(13,46)(14,42)(15,47)(17,48)(18,49)(19,52)(20,50)(22,33)(25,29)(27,58)(28,59)(30,57)(31,53)(32,56)(36,61)(37,62)(38,54)(39,45)(41,44)(43,64)(51,55)(60,63)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,f^c=fi,g^c=gh,g^d=gi,g^e=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, 54942);