#147231

order	512
nilpotency class	4
isoclinism	isoclinic class of #12948
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10885
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dh,f^b=fi,h^b=hi,e^c=ei

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, 54427);