#147433

order	512
nilpotency class	4
isoclinism	isoclinic class of #13247
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1972
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10927
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

685 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)(3,15)(4,22,26,43)(5,28)(6,12)(7,32)(8,14)(9,17)(10,25,39,27)(11,30)(13,29)(16,47,50,64)(18,45)(19,36)(20,55)(21,46)(23,59,53,44)(24,40,33,38)(31,41)(34,60,56,62)(35,51)(37,49,63,54)(42,52)(48,58,61,57);(1,3,7,19,29,18,30,52,8,21,5,17,6,20,31,51)(2,9,11,15,12,42,32,55,14,36,41,46,13,35,28,45)(4,23,24,60,25,16,48,49,26,53,33,62,27,50,61,54)(10,37,40,47,43,34,57,59,39,63,38,64,22,56,58,44);(1,4,6,27,8,26,29,25)(2,10,12,43,14,39,13,22)(3,16,18,53,21,50,20,23)(5,24,30,61,7,33,31,48)(9,34,35,37,36,56,42,63)(11,38,32,58,41,40,28,57)(15,44,45,64,46,59,55,47)(17,49,51,62,19,54,52,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54629);