#496130

order	512
nilpotency class	4
isoclinism	isoclinic class of #496130
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21697
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33720	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

659 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)(3,14)(4,20,23,34)(5,24)(6,11)(7,13)(8,17)(9,22,33,31)(10,15)(12,25)(16,44,48,63)(18,38)(19,40)(21,32,52,60)(26,35)(27,54,56,61)(28,41)(29,58)(30,50)(36,43)(37,47,62,55)(39,49)(42,51,59,53)(45,64,57,46);(1,3,15,41,6,18,26,50,7,19,43,58,25,49,5,17)(2,8,24,39,12,29,36,40,13,30,35,38,11,28,10,14)(4,16,42,54,22,47,52,64,23,48,59,61,31,55,21,46)(9,27,53,63,34,57,60,62,33,56,51,44,20,45,32,37);(1,4,7,23)(2,9)(3,16,19,48)(5,21,26,52)(6,22,25,31)(8,27)(10,32)(11,20)(12,34)(13,33)(14,37)(15,42,43,59)(17,46,50,64)(18,47,49,55)(24,53)(28,45)(29,57)(30,56)(35,51)(36,60)(38,44)(39,63)(40,62)(41,54,58,61);(1,2)(3,8)(4,20,23,34)(5,10)(6,12)(7,13)(9,31,33,22)(11,25)(14,17)(15,24)(16,45,48,57)(18,29)(19,30)(21,51,52,53)(26,36)(27,55,56,47)(28,49)(32,59,60,42)(35,43)(37,61,62,54)(38,58)(39,41)(40,50)(44,64,63,46)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=fh,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403326);