#419464

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15270
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1299 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)(3,15,18,9)(4,10,23,36)(5,25)(6,14)(7,13)(8,12)(11,26)(16,30)(17,45,46,56)(19,32,20,33)(21,50)(22,52,53,61)(24,40,28,37)(27,41)(29,39)(31,47,38,49)(34,48)(35,54,60,55)(42,44)(43,59)(51,62)(57,63)(58,64);(1,3)(2,9)(4,21)(5,17)(6,20)(7,19)(8,18)(10,34)(11,38)(12,15)(13,33)(14,32)(16,24)(22,51)(23,48)(25,56)(26,49)(27,46)(28,44)(29,47)(30,37)(31,39)(35,59)(36,50)(40,42)(41,45)(43,54)(52,57)(53,63)(55,64)(58,60)(61,62);(1,4)(2,10,12,36)(3,16)(5,22)(6,28)(7,24)(8,23)(9,30,15,42)(11,35,39,60)(13,40,14,37)(17,43)(18,44)(19,48)(20,21)(25,52,41,61)(26,55)(27,53)(29,54)(31,57,38,62)(32,50,33,34)(45,58,56,59)(46,64)(47,63)(49,51);(1,5)(2,11)(3,17)(4,22)(6,26)(7,29)(8,27)(9,31)(10,35)(12,39)(13,25)(14,41)(15,38)(16,43)(18,46)(19,47)(20,49)(21,51)(23,53)(24,54)(28,55)(30,57)(32,56)(33,45)(34,58)(36,60)(37,61)(40,52)(42,62)(44,64)(48,63)(50,59)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326660);