#420397

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15277
prime radical	#60469	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1279 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=e,e^2=i,f^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,f^c=fi,g^c=gi,h^c=hi,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, 327593);