#502381

order	512
nilpotency class	4
isoclinism	isoclinic class of #500737
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3325
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#25321
prime radical	#33594	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409577);