#493132

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1770
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285672, #72285848, #72285871, …
maximal normal subgroup of	#72285672, #72285848, #72285871, …
maximal quotient of	#72342630

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400328);