#482034

order	512
nilpotency class	4
isoclinism	isoclinic class of #482033
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1519
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#60462	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

980 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)(3,14,17,41)(4,20,21,10)(5,25,27,39)(6,12)(7,30)(8,13)(9,16)(11,28,37,29)(15,42)(18,36,19,49)(22,54)(23,52,24,35)(26,38)(31,44,50,63)(32,47)(33,48)(34,56)(40,43)(45,62)(46,60)(51,57)(53,58)(55,59,64,61);(1,3,8,17)(2,9,13,33)(4,15,21,43)(5,26,27,47)(6,19,7,18)(10,31,20,50)(11,36,37,49)(12,38,30,32)(14,39,41,25)(16,28,48,29)(22,55,53,64)(23,46,24,45)(34,60,51,62)(35,61,52,59)(40,54,42,58)(44,56,63,57);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,21)(9,31)(11,34)(12,35)(13,20)(14,40)(16,44)(17,43)(18,45)(19,46)(25,58)(26,55)(27,53)(28,56)(29,57)(30,52)(32,59)(33,50)(36,60)(37,51)(38,61)(39,54)(41,42)(47,64)(48,63)(49,62);(4,21)(10,20)(15,43)(22,53)(23,24)(31,50)(34,51)(35,52)(40,42)(44,63)(45,46)(54,58)(55,64)(56,57)(59,61)(60,62)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cgh,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^e=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389230);