#238419

order	512
nilpotency class	3
isoclinism	isoclinic class of #238414
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145615);