#240604

order	512
nilpotency class	3
isoclinism	isoclinic class of #240595
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

786 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)(4,16)(5,10)(6,21)(7,11)(8,14)(9,19)(13,30)(15,24)(17,29)(18,28)(20,27)(22,25)(23,31)(26,32)(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)(4,17)(5,14)(6,22)(7,15)(9,26)(10,24)(11,25)(12,21)(13,19)(16,23)(18,32)(20,31)(27,30)(28,29)(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);(1,4,6,18,7,19,5,20)(2,9,10,28,11,16,21,27)(3,13,14,32,15,17,22,31)(8,23,12,29,25,26,24,30)(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)(2,10,11,21)(3,14,15,22)(4,18,19,20)(8,24,25,12)(9,27,16,28)(13,31,17,32)(23,29,26,30)(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=gi,c^b=cg,e^b=eh,f^b=fi,d^c=dhi,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147800);