#238069

order	512
nilpotency class	3
isoclinism	isoclinic class of #238065
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

726 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,d^c=di,e^c=eh,f^c=fh,g^c=gh

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, 145265);