#427985

order	512
nilpotency class	3
isoclinism	isoclinic class of #417630
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15376
prime radical	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1234 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 335181);