#503498

order	512
nilpotency class	4
isoclinism	isoclinic class of #503497
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25900
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3540	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

739 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)(3,15)(4,19,24,50)(5,9)(6,27)(7,13)(8,14)(10,23,37,58)(11,18)(12,28)(16,43,45,48)(17,42)(20,51)(21,47,55,63)(22,56,57,49)(25,32)(26,33)(29,38)(30,40)(31,54,61,64)(34,46)(35,60,41,44)(36,62,59,53)(39,52);(1,3,6,5,8,18,28,26)(2,9,12,15,14,33,27,11)(4,16,22,21,24,45,57,55)(7,29,20,52,30,17,46,25)(10,31,36,41,37,61,59,35)(13,39,34,38,40,32,51,42)(19,47,49,43,50,63,56,48)(23,44,53,54,58,60,62,64);(1,4,20,53)(2,10,34,49)(3,16,25,54)(5,21,29,60)(6,22,30,58)(7,23,28,57)(8,24,46,62)(9,31,42,43)(11,35,32,47)(12,36,40,50)(13,19,27,59)(14,37,51,56)(15,41,39,63)(17,44,26,55)(18,45,52,64)(33,61,38,48);(1,5,6,18,8,26,28,3)(2,11,12,9,14,15,27,33)(4,21,22,45,24,55,57,16)(7,25,20,29,30,52,46,17)(10,35,36,31,37,41,59,61)(13,38,34,32,40,42,51,39)(19,48,49,47,50,43,56,63)(23,54,53,60,58,64,62,44)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410694);