#480231

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=g,f^2=h,h^2=i,b^a=be,e^a=eg,c^b=ci,f^b=fg,d^c=dgi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387427);