#458075

order	512
nilpotency class	3
isoclinism	isoclinic class of #455563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16309
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

893 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)(4,10)(5,23)(6,14)(7,13)(8,12)(9,18)(11,26)(16,42)(17,31)(19,41)(20,37)(21,40)(22,36)(24,34)(25,39)(27,38)(28,35)(29,32)(30,46)(33,45)(43,49)(44,55)(47,56)(48,62)(50,59)(51,58)(52,61)(53,54)(57,64)(60,63);(1,3,8,29,7,19,6,18)(2,9,14,41,13,32,12,15)(4,16,21,45,22,47,28,46)(5,17,25,49,27,50,26,51)(10,33,35,42,36,30,40,56)(11,31,38,58,39,59,23,43)(20,44,53,64,52,63,24,48)(34,60,61,57,54,55,37,62);(1,4,8,21,7,22,6,28)(2,10,14,35,13,36,12,40)(3,16,29,45,19,47,18,46)(5,24,25,20,27,53,26,52)(9,30,41,56,32,33,15,42)(11,37,38,34,39,61,23,54)(17,48,49,44,50,64,51,63)(31,57,58,55,59,62,43,60);(1,5)(2,11)(3,17)(4,20)(6,26)(7,27)(8,25)(9,31)(10,34)(12,23)(13,39)(14,38)(15,43)(16,44)(18,51)(19,50)(21,53)(22,52)(24,28)(29,49)(30,55)(32,59)(33,60)(35,61)(36,54)(37,40)(41,58)(42,57)(45,64)(46,48)(47,63)(56,62)

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,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, 365271);