#430346

order	512
nilpotency class	3
isoclinism	isoclinic class of #430345
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15418
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1129 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337542);