#148233

order	512
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2067
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11076
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

671 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=ghi,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, 55429);