#148532

order	512
nilpotency class	4
isoclinism	isoclinic class of #140203
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1934
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11189
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

643 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)(3,15,20,35)(4,22,26,54)(5,12)(6,30)(7,32)(8,14)(9,17,37,18)(10,24,41,57)(11,28)(13,31)(16,47)(19,45,52,38)(21,43,33,36)(23,39)(25,59,58,42)(27,61,60,40)(29,44)(34,55)(46,48)(49,62)(50,63)(51,53)(56,64);(1,3)(2,9)(4,23)(5,19)(6,17)(7,33)(8,20)(10,39)(11,43)(12,35)(13,45)(14,37)(15,44)(16,25)(18,28)(21,31)(22,53)(24,55)(26,56)(27,49)(29,52)(30,36)(32,38)(34,40)(41,64)(42,46)(47,54)(48,60)(50,57)(51,58)(59,62)(61,63);(1,4,5,24,7,27,28,58,8,26,29,57,31,60,6,25)(2,10,11,22,13,42,12,40,14,41,30,54,32,59,44,61)(3,16,17,48,21,50,52,56,20,51,18,49,33,55,19,23)(9,34,15,46,38,53,36,39,37,63,35,62,45,47,43,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307185, #72307242, #72307253, …
maximal normal subgroup of	#72307185, #72307242, #72307253, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55728);