#471396

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16795
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

640 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)(3,9,17,32)(4,19)(5,11,26,38)(6,14)(7,13)(8,12)(10,22)(15,33,43,61)(16,30)(18,40,27,31)(20,29,50,56)(21,51,53,48)(23,35)(24,36)(25,41,28,39)(34,52,37,54)(42,62)(44,60)(45,59)(46,58)(47,57)(49,64)(55,63);(1,3,7,18,8,17,6,27)(2,9,14,31,12,32,13,40)(4,20,22,15,24,50,23,43)(5,16,28,46,26,45,25,47)(10,33,19,29,35,61,36,56)(11,30,41,57,38,59,39,58)(21,49,52,42,53,63,54,44)(34,60,48,55,37,62,51,64);(1,4,6,22,8,24,7,23)(2,10,13,19,12,35,14,36)(3,15,18,20,17,43,27,50)(5,21,25,52,26,53,28,54)(9,29,31,33,32,56,40,61)(11,37,39,51,38,34,41,48)(16,44,46,63,45,42,47,49)(30,55,57,60,59,64,58,62);(1,5,6,25,8,26,7,28)(2,11,14,41,12,38,13,39)(3,16,27,47,17,45,18,46)(4,21,22,52,24,53,23,54)(9,30,31,57,32,59,40,58)(10,34,36,51,35,37,19,48)(15,42,50,63,43,44,20,49)(29,55,33,60,56,64,61,62)

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=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, 378592);