#451215

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15989
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

748 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=ef,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 358411);