#471132

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16760
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

515 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=efh,c^2=e,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, 378328);