#478821

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17170
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:

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

Polycyclic presentation:

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

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, 386017);