#494310

order	512
nilpotency class	4
isoclinism	isoclinic class of #33531
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1862
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1027 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=di,e^b=eh,f^b=fhi,h^b=hi,e^c=ehi,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401506);