#478744

order	512
nilpotency class	3
isoclinism	isoclinic class of #477835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17165
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

645 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)(5,27,32,11)(6,13)(7,12)(8,14)(9,18)(10,25)(16,47)(17,46,28,58)(19,36)(20,35)(22,37)(23,54,57,42)(24,40)(26,39)(29,59,56,38)(30,43,31,44)(33,50)(34,52,41,53)(45,49)(48,63,64,60)(51,61,55,62);(1,3,6,18,8,20,7,19)(2,9,12,15,14,36,13,35)(4,22,24,16,26,49,25,50)(5,28,30,53,32,17,31,52)(10,37,39,33,40,45,21,47)(11,41,43,46,27,34,44,58)(23,55,29,48,57,51,56,64)(38,62,42,60,59,61,54,63);(1,4,7,25,8,26,6,24)(2,10,13,21,14,40,12,39)(3,16,18,49,20,50,19,22)(5,29,31,23,32,56,30,57)(9,33,15,45,36,47,35,37)(11,42,44,38,27,54,43,59)(17,51,52,64,28,55,53,48)(34,61,58,63,41,62,46,60);(1,5)(2,11)(3,17)(4,23)(6,31)(7,30)(8,32)(9,34)(10,38)(12,44)(13,43)(14,27)(15,46)(16,48)(18,53)(19,52)(20,28)(21,42)(22,51)(24,56)(25,29)(26,57)(33,60)(35,58)(36,41)(37,61)(39,54)(40,59)(45,62)(47,63)(49,55)(50,64)

Polycyclic presentation:

2^9|a^2=fg,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,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^d=ei,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385940);