#499119

order	512
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2044
commutator, Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#2044
prime radical	#60749	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1076 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)(3,14)(4,19)(5,11)(6,26)(7,12)(8,13)(9,16)(10,23)(15,45)(17,44)(18,33)(20,53)(21,52)(22,56)(24,41)(25,39)(27,35)(28,51)(29,60)(30,38)(31,42)(32,61)(34,48)(36,64)(37,55)(40,58)(43,47)(46,63)(49,62)(50,54)(57,59);(1,3)(2,9)(4,20)(5,16)(6,27)(7,29)(8,18)(10,36)(11,33)(12,40)(13,35)(14,26)(15,24)(17,31)(19,50)(21,54)(22,46)(23,49)(25,58)(28,48)(30,47)(32,38)(34,42)(37,43)(39,44)(41,63)(45,52)(51,60)(53,56)(55,64)(57,61)(59,62);(1,4,6,22,8,24,5,21)(2,10,11,37,13,38,26,59)(3,15,16,46,18,20,27,54)(7,30,28,55,31,23,25,57)(9,32,14,43,35,36,33,62)(12,41,39,56,42,19,51,52)(17,49,48,64,29,47,58,61)(34,50,44,53,40,63,60,45);(1,5,8,6)(2,11,13,26)(3,16,18,27)(4,21,24,22)(7,25,31,28)(9,33,35,14)(10,37,38,59)(12,39,42,51)(15,46,20,54)(17,48,29,58)(19,52,41,56)(23,55,30,57)(32,62,36,43)(34,60,40,44)(45,63,53,50)(47,61,49,64)

Polycyclic presentation:

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

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, 406315);