#457987

order	512
nilpotency class	3
isoclinism	isoclinic class of #454109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16307
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

760 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,h^c=hi,e^d=ei,f^d=fi,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, 365183);