#430486

order	512
nilpotency class	3
isoclinism	isoclinic class of #417630
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15420
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1151 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)(3,15,19,34)(4,10)(5,26,29,40)(6,12)(7,13)(8,14)(9,18,33,20)(11,30,41,28)(16,44,47,35)(17,48)(21,51,46,31)(22,53,56,58)(23,37)(24,42)(25,38)(27,43)(32,50)(36,55,61,57)(39,49)(45,63)(52,60)(54,59)(62,64);(1,3,6,20,8,19,7,18)(2,9,12,34,14,33,13,15)(4,21,23,47,25,46,24,16)(5,27,30,50,29,49,28,17)(10,35,37,31,38,44,42,51)(11,39,26,48,41,43,40,32)(22,54,57,64,56,45,55,52)(36,59,58,62,61,63,53,60);(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=f,b^2=ef,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, 337682);