#496503

order	512
nilpotency class	4
isoclinism	isoclinic class of #496502
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21865
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

734 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=ce,f^b=fi,g^b=gi,f^c=fh,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403699);