#240726

order	512
nilpotency class	3
isoclinism	isoclinic class of #240721
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

772 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)(3,12)(4,17)(5,10)(6,22)(7,11)(8,15)(9,21)(13,27)(14,24)(16,26)(18,32)(19,31)(20,25)(23,29)(28,30)(33,34)(35,39)(36,46)(37,49)(38,42)(40,48)(41,52)(43,60)(44,58)(45,55)(47,59)(50,54)(51,57)(53,62)(56,63)(61,64);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,23)(10,26)(11,24)(12,22)(13,21)(17,30)(19,28)(20,29)(25,32)(27,31)(33,35,38,45)(34,39,42,55)(36,43,47,61)(37,50,52,44)(40,56,57,53)(41,58,49,54)(46,60,59,64)(48,63,51,62);(1,4,6,20,7,21,5,19)(2,9,10,25,11,17,22,31)(3,13,15,29,16,18,14,28)(8,23,12,27,24,30,26,32)(33,36)(34,40)(35,43)(37,51)(38,47)(39,53)(41,59)(42,57)(44,63)(45,61)(46,49)(48,52)(50,62)(54,64)(55,56)(58,60);(1,5,7,6)(2,10,11,22)(3,14,16,15)(4,19,21,20)(8,12,24,26)(9,25,17,31)(13,28,18,29)(23,27,30,32)(36,47)(40,57)(43,61)(46,59)(48,51)(53,56)(60,64)(62,63)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fhi,d^c=di,f^c=fhi,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, 147922);