#456738

order	512
nilpotency class	3
isoclinism	isoclinic class of #455393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16285
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

940 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:

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

Polycyclic presentation:

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

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, 363934);