#118239

order	512
nilpotency class	3
isoclinism	isoclinic class of #118234
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

539 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25435);