#273783

order	512
nilpotency class	3
isoclinism	isoclinic class of #273755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3515	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=i,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fh,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180979);