#424970

order	512
nilpotency class	3
isoclinism	isoclinic class of #420055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2476
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15326
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

943 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)(3,15)(4,21)(5,27,30,11)(6,13)(7,12)(8,14)(9,18)(10,25)(16,32)(17,40,48,43)(19,35)(20,34)(22,36)(23,51,55,49)(24,39)(26,38)(28,33,47,60)(29,41,31,42)(37,53,56,54)(44,63,52,57)(45,58)(46,59)(50,62,64,61);(1,3,6,18,8,20,7,19)(2,9,12,15,14,35,13,34)(4,22,24,16,26,45,25,46)(5,28,29,48,30,47,31,17)(10,36,38,59,39,58,21,32)(11,40,41,60,27,43,42,33)(23,52,53,64,55,44,54,50)(37,57,49,62,56,63,51,61);(1,4,7,25,8,26,6,24)(2,10,13,21,14,39,12,38)(3,16,19,22,20,46,18,45)(5,23,29,53,30,55,31,54)(9,32,34,58,35,59,15,36)(11,37,41,49,27,56,42,51)(17,44,28,50,48,52,47,64)(33,57,40,62,60,63,43,61);(1,5)(2,11)(3,17)(4,23)(6,31)(7,29)(8,30)(9,33)(10,37)(12,42)(13,41)(14,27)(15,43)(16,44)(18,47)(19,28)(20,48)(21,49)(22,50)(24,54)(25,53)(26,55)(32,57)(34,40)(35,60)(36,61)(38,51)(39,56)(45,64)(46,52)(58,62)(59,63)

Polycyclic presentation:

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

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, 332166);