#493202

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#20476
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

519 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320434, #72320704, #72320755
maximal normal subgroup of	#72320434, #72320704, #72320755

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400398);