#482711

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	#60717	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

676 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)(3,15,18,45)(4,21,22,35)(5,26,28,36)(6,13)(7,10)(8,14)(9,17)(11,24,38,25)(12,30,41,31)(16,46)(19,39,20,43)(23,40)(27,42)(29,37)(32,53)(33,56,63,60)(34,51)(44,50)(47,64)(48,54,52,61)(49,62)(55,58)(57,59);(1,3)(2,9)(4,16)(5,27)(6,20)(7,19)(8,18)(10,32)(11,33)(12,39)(13,42)(14,34)(15,26)(17,31)(21,54)(22,47)(23,56)(24,50)(25,49)(28,53)(29,60)(30,51)(35,61)(36,45)(37,64)(38,63)(40,46)(41,43)(44,59)(48,58)(52,57)(55,62);(2,10,14,13)(4,22)(5,28)(9,32,34,42)(11,21,38,35)(12,26,41,36)(15,43,45,39)(16,47)(17,51)(24,25)(27,53)(30,31)(33,54,63,61)(37,59,40,55)(44,46,62,64)(49,50);(1,4)(2,11,14,38)(3,16)(5,29)(6,24)(7,25)(8,22)(9,33,34,63)(10,35,13,21)(12,40,41,37)(15,44,45,62)(17,52)(18,47)(19,49)(20,50)(23,28)(26,59,36,55)(27,60)(30,58)(31,57)(32,61,42,54)(39,46,43,64)(48,51)(53,56)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,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, 389907);