#383746

order	512
nilpotency class	3
isoclinism	isoclinic class of #383745
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14363
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1315 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,f^d=fi,h^d=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286106, #72286213, #72286267
maximal normal subgroup of	#72286106, #72286213, #72286267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290942);