#153582

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#117
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₃₂	C₂³
socle	C₂	#13640
prime radical	C₂³	M₆₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ #1397 ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

106 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=d,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ch,d^a=di,e^b=ei,e^c=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225395, #72225404, #72225405, …
maximal normal subgroup of	#72225395, #72225404, #72225405, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60778);