#147679

order	512
nilpotency class	4
isoclinism	isoclinic class of #13389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10969
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1005 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)(3,14,18,43)(4,19,23,9)(5,25)(6,12)(7,11)(8,16,32,51)(10,27)(13,30)(15,45)(17,38,29,44)(20,54)(21,56,59,61)(22,41,24,37)(26,33,50,34)(28,40)(31,46)(35,55)(36,60,53,58)(39,49)(42,48)(47,63)(52,64)(57,62);(1,3)(2,8)(4,20)(5,26)(6,29)(7,18)(9,35)(10,38)(11,32)(12,34)(13,33)(14,40)(15,22)(16,49)(17,30)(19,52)(21,57)(23,47)(24,48)(25,43)(27,51)(28,50)(31,37)(36,42)(39,44)(41,62)(45,53)(46,59)(54,61)(55,58)(56,63)(60,64);(1,4)(2,9)(3,15)(5,21)(6,24)(7,23)(8,31)(10,36)(11,19)(12,41)(13,37)(14,42)(16,46)(17,47)(18,48)(20,29)(22,30)(25,61)(26,55)(27,60)(28,59)(32,62)(33,52)(34,35)(38,54)(39,53)(40,56)(43,45)(44,63)(49,58)(50,64)(51,57)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54875);