#419798

order	512
nilpotency class	3
isoclinism	isoclinic class of #419798
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15273
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1534 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264921, #72264949, #72265017
maximal normal subgroup of	#72264921, #72264949, #72265017
maximal quotient of	#72314082

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326994);