#502100

order	512
nilpotency class	4
isoclinism	isoclinic class of #502097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25076
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33539	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

752 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409296);