#147612

order	512
nilpotency class	4
isoclinism	isoclinic class of #137786
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1899
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10966
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

870 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)(4,20,24,40)(5,25,28,11)(6,12)(7,13)(8,14)(9,17,33,49)(10,23,36,30)(16,44,47,62)(18,42)(19,43)(21,41,46,63)(22,52)(26,38,27,37)(29,58)(31,59)(32,48,39,56)(34,45)(35,53)(50,54)(51,61)(55,57)(60,64);(1,3)(2,9)(4,21)(5,17)(6,18)(7,29)(8,19)(10,34)(11,15)(12,32)(13,39)(14,33)(16,23)(20,31)(22,51)(24,46)(25,43)(26,48)(27,56)(28,49)(30,47)(35,62)(36,60)(37,42)(38,58)(40,61)(41,50)(44,55)(45,53)(52,63)(54,59)(57,64);(1,4)(2,10,14,36)(3,16)(5,22)(6,23)(7,30)(8,24)(9,31,33,61)(11,35,25,55)(12,20,13,40)(15,41,43,63)(17,45)(18,46)(19,47)(21,29)(26,53)(27,57)(28,54)(32,60,39,34)(37,50,38,52)(42,44,58,62)(48,59)(49,64)(51,56)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54808);