#469879

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16666
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

793 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)(3,13,16,8)(4,9)(5,22)(6,26)(7,11)(10,25)(12,29)(14,30,18,39)(15,43,45,57)(17,32,27,37)(19,48)(20,34)(21,38)(23,50)(24,51)(28,35)(31,46,40,54)(33,49)(36,53)(41,61,44,63)(42,58,47,56)(52,59)(55,64,60,62);(1,3)(2,8)(4,18)(5,15)(6,27)(7,16)(9,30)(10,31)(11,13)(12,37)(14,20)(17,29)(19,44)(21,42)(22,43)(23,41)(24,45)(25,54)(26,32)(28,47)(33,55)(34,39)(35,58)(36,40)(38,56)(46,53)(48,63)(49,62)(50,61)(51,57)(52,64)(59,60);(1,4)(2,9)(3,14,16,18)(5,23)(6,28)(7,20)(8,30,13,39)(10,33)(11,34)(12,38)(15,44,45,41)(17,47,27,42)(19,24)(21,29)(22,50)(25,49)(26,35)(31,55,40,60)(32,58,37,56)(36,59)(43,61,57,63)(46,62,54,64)(48,51)(52,53);(1,5)(2,10,11,36)(3,15)(4,19,20,23)(6,25)(7,24)(8,31,13,40)(9,33)(12,22,26,51)(14,41,18,44)(16,45)(17,46)(21,49,28,52)(27,54)(29,53)(30,55)(32,57,37,43)(34,59)(35,48)(38,50)(39,60)(42,62,47,64)(56,61)(58,63)

Polycyclic presentation:

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

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, 377075);