#429426

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15402
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

730 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=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, 336622);