#147661

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
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)(3,14,18,42)(4,19,22,40)(5,24)(6,11)(7,12)(8,16,32,51)(9,21,36,23)(10,26)(13,29)(15,44)(17,37,28,43)(20,54,57,35)(25,31,50,33)(27,38)(30,55)(34,60)(39,49)(41,47)(45,59)(46,61)(48,62)(52,64)(53,56,63,58);(1,3)(2,8)(4,15)(5,25)(6,28)(7,18)(9,34)(10,37)(11,33)(12,32)(13,31)(14,38)(16,49)(17,29)(19,52)(20,55)(21,47)(22,48)(23,46)(24,42)(26,51)(27,50)(30,36)(35,44)(39,43)(40,59)(41,53)(45,56)(54,62)(57,60)(58,64)(61,63);(1,4)(2,9)(3,15)(5,20)(6,23)(7,22)(8,30)(10,35)(11,40)(12,36)(13,19)(14,41)(16,45)(17,47)(18,48)(21,29)(24,53)(25,55)(26,58)(27,57)(28,46)(31,59)(32,34)(33,52)(37,62)(38,63)(39,54)(42,61)(43,44)(49,56)(50,60)(51,64)

Polycyclic presentation:

2^9|a^2=g,d^2=h,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=dhi,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279962, #72280034, #72307169, …
maximal normal subgroup of	#72279962, #72280034, #72307169, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54857);