#273786

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	#3518	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)(3,10,16,26)(4,9)(5,8,15,22)(6,11)(7,12)(13,29,30,23)(14,28)(17,32,31,21)(18,24)(19,25)(20,27)(33,34,38,43,39,37,56,42)(35,44,48,46,49,58,47,40)(36,50,54,52,55,60,53,41)(45,57,62,64,63,61,51,59);(1,3,7,16)(2,8,12,22)(4,13,19,30)(5,14,15,6)(9,21,25,32)(10,11,26,28)(17,18,31,20)(23,27,29,24)(33,35)(34,40)(36,51)(37,46)(38,48)(39,49)(41,59)(42,58)(43,44)(45,54)(47,56)(50,57)(52,64)(53,63)(55,62)(60,61);(1,4,7,19)(2,9,12,25)(3,13,16,30)(5,17,15,31)(6,20,14,18)(8,21,22,32)(10,23,26,29)(11,27,28,24)(33,36)(34,41)(35,45)(37,52)(38,53)(39,55)(40,57)(42,50)(43,60)(44,59)(46,61)(47,62)(48,51)(49,63)(54,56)(58,64);(1,5)(2,10)(3,14)(4,17)(6,16)(7,15)(8,11)(9,23)(12,26)(13,18)(19,31)(20,30)(21,27)(22,28)(24,32)(25,29)(33,37,39,34)(35,46,49,40)(36,52,55,41)(38,42,56,43)(44,48,58,47)(45,61,63,57)(50,54,60,53)(51,64,62,59)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=hi,d^2=h,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, 180982);