#148733

order	512
nilpotency class	4
isoclinism	isoclinic class of #136628
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11208
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 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)(3,14,18,33)(4,19)(5,10)(6,27)(7,13)(8,16,32,48)(9,22)(11,25)(12,26)(15,43,46,51)(17,31,50,62)(20,54,44,64)(21,39)(23,52)(24,53)(28,36)(29,38)(30,60,61,49)(34,45,41,59)(35,55)(37,56)(40,57)(42,63,58,47);(1,3,7,18)(2,8)(4,20)(5,16,26,48)(6,17,29,50)(9,34,37,41)(10,14)(11,31)(12,33)(13,32)(15,21)(19,51,53,43)(22,49)(23,46)(24,44)(25,47,57,63)(27,58)(28,59)(30,36,61,35)(38,42)(39,54,52,64)(40,62)(45,55)(56,60);(1,4,5,21,7,24,26,23)(2,9,10,35,13,37,12,36)(3,15,16,44,18,46,48,20)(6,28,25,22,29,55,57,56)(8,30,14,41,32,61,33,34)(11,39,38,53,40,52,27,19)(17,49,47,45,50,60,63,59)(31,43,42,64,62,51,58,54)

Polycyclic presentation:

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

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, 55929);