#147654

order	512
nilpotency class	4
isoclinism	isoclinic class of #13318
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10969
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1019 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)(4,19,22,39)(5,23)(6,12)(7,30)(8,13)(9,16)(10,21,38,29)(11,27)(15,46,49,64)(17,44)(18,45)(20,37,25,41)(24,33)(26,42)(28,57)(31,40)(32,55,36,58)(34,52)(35,59)(43,48,61,50)(47,53,51,62)(54,60,63,56);(1,3)(2,9)(4,15)(5,24)(6,17)(7,31)(8,18)(10,36)(11,40)(12,33)(13,34)(14,23)(16,27)(19,53)(20,55)(21,48)(22,49)(25,58)(26,52)(28,59)(29,50)(30,35)(32,38)(37,46)(39,62)(41,64)(42,44)(43,63)(45,57)(47,56)(51,60)(54,61);(1,4)(2,10)(3,15)(5,25)(6,29)(7,21)(8,22)(9,32)(11,41)(12,39)(13,38)(14,43)(16,51)(17,50)(18,49)(19,30)(20,28)(23,54)(24,58)(26,56)(27,60)(31,48)(33,53)(34,36)(35,62)(37,42)(40,46)(44,64)(45,61)(47,52)(55,59)(57,63)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279962, #72280256, #72307182, …
maximal normal subgroup of	#72279962, #72280256, #72307182, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54850);