#480124

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#99
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1459
prime radical	#3449	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

967 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387320);