#493923

order	512
nilpotency class	4
isoclinism	isoclinic class of #33657
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1824
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

971 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)(3,9,17,34,15,32,18,50)(4,19,22,38,20,52,23,56)(5,24,27,39,29,58,28,11)(10,26,36,21,35,55,40,53)(14,41,44,62,42,51,45,64)(16,46,48,37,25,57,49,33)(31,47,61,43,60,59,63,54);(1,3)(2,9)(4,14)(5,25)(6,17)(7,18)(8,15)(10,31)(11,37)(12,34)(13,32)(16,29)(19,51)(20,42)(21,54)(22,44)(23,45)(24,57)(26,59)(27,49)(28,48)(30,50)(33,39)(35,60)(36,61)(38,64)(40,63)(41,52)(43,53)(46,58)(47,55)(56,62);(1,4,7,22,8,20,6,23)(2,10,12,40,13,35,30,36)(3,14,18,44,15,42,17,45)(5,26,28,21,29,55,27,53)(9,31,34,63,32,60,50,61)(11,38,24,19,39,56,58,52)(16,47,49,43,25,59,48,54)(33,62,46,41,37,64,57,51);(3,15)(9,32)(14,42)(16,25)(17,18)(31,60)(33,37)(34,50)(41,51)(43,54)(44,45)(46,57)(47,59)(48,49)(61,63)(62,64)

Polycyclic presentation:

2^9|a^2=g,c^2=eh,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fh,g^c=gi,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, 401119);