#493227

order	512
nilpotency class	3
isoclinism	isoclinic class of #356478
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	D₄×#97
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#347 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,h^2=i,b^a=be,c^a=chi,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=dh,f^c=fi,g^d=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285481, #72285521, #72285672
maximal normal subgroup of	#72285481, #72285521, #72285672
maximal quotient of	#72287262, D₈×#347, #72309489, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400423);