#482713

order	512
nilpotency class	4
isoclinism	isoclinic class of #482711
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17595
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

666 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)(4,19,20,34)(5,23,25,38)(6,9)(7,13)(8,16,32,49)(10,22,36,53)(11,28,40,48)(12,27)(15,44,45,64)(17,41)(18,43)(21,39)(24,30,58,33)(26,35)(29,37)(31,59)(42,47,62,56)(46,61)(50,51)(52,60)(54,63)(55,57);(1,3,7,18)(2,8,13,32)(4,15,20,45)(5,24,25,58)(6,29,27,17)(9,30,12,33)(10,31,36,63)(11,37,40,41)(14,23,43,38)(16,48,49,28)(19,51,34,61)(21,54,26,59)(22,56,53,47)(35,44,39,64)(42,57,62,52)(46,60,50,55);(2,9,13,12)(4,20)(5,25)(8,30,32,33)(10,34,36,19)(11,38,40,23)(14,41,43,37)(15,45)(16,49)(22,53)(24,58)(28,48)(31,61,63,51)(35,52,39,57)(42,64,62,44)(47,56);(1,4,7,20)(2,10)(3,15,18,45)(5,26,25,21)(6,22,27,53)(8,31)(9,19)(11,39)(12,34)(13,36)(14,42)(16,50,49,46)(17,47,29,56)(23,57)(24,59,58,54)(28,60,48,55)(30,51)(32,63)(33,61)(35,40)(37,64)(38,52)(41,44)(43,62)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=gh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei,f^d=fi,f^e=fi

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, 389909);