#482040

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

972 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=hi,f^2=i,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, 389236);