#235497

order	512
nilpotency class	3
isoclinism	isoclinic class of #235492
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60970	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1146 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)(3,17)(4,22)(5,12)(6,30)(7,33)(8,16)(9,15)(10,18)(11,25)(13,29)(14,49)(19,54)(20,38)(21,52)(23,46)(24,56)(26,40)(27,59)(28,47)(31,41)(32,37)(34,44)(35,50)(36,51)(39,57)(42,61)(43,58)(45,55)(48,53)(60,63)(62,64);(1,3)(2,10)(4,23)(5,26)(6,31)(7,34)(8,20)(9,21)(11,42)(12,22)(13,38)(14,43)(15,40)(16,41)(17,30)(18,29)(19,35)(24,28)(25,36)(27,57)(32,53)(33,63)(37,44)(39,51)(45,59)(46,56)(47,52)(48,50)(49,55)(54,64)(58,61)(60,62);(1,4,8,21)(2,11,16,45)(3,9,20,23)(5,26,28,24)(6,31,29,18)(7,19,35,34)(10,37,41,64)(12,43,47,39)(13,48,30,63)(14,17,51,38)(15,44,46,54)(22,50,52,33)(25,36,55,49)(27,60,61,53)(32,58,62,57)(40,42,56,59);(1,5,8,28)(2,12,16,47)(3,18,20,31)(4,24,21,26)(6,9,29,23)(7,27,35,61)(10,38,41,17)(11,39,45,43)(13,46,30,15)(14,37,51,64)(19,53,34,60)(22,56,52,40)(25,57,55,58)(32,36,62,49)(33,59,50,42)(44,63,54,48)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,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, 142693);