#471078

order	512
nilpotency class	3
isoclinism	isoclinic class of #460970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2759
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16747
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

616 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:

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

Polycyclic presentation:

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

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, 378274);