#273775

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	#3517	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

751 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:

(1,2,5,11,7,8,6,12)(3,13,17,22,16,24,14,9)(4,18,19,26,20,10,21,25)(15,27,28,23,29,32,30,31)(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);(2,8)(3,14)(5,6)(9,13)(10,18)(15,28)(16,17)(19,21)(22,24)(23,27)(29,30)(31,32)(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);(1,3)(2,9)(4,15)(5,14)(6,17)(7,16)(8,22)(10,23)(11,24)(12,13)(18,31)(19,30)(20,29)(21,28)(25,27)(26,32)(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);(1,4)(2,10)(3,15)(5,21)(6,19)(7,20)(8,18)(9,23)(11,26)(12,25)(13,27)(14,28)(16,29)(17,30)(22,31)(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=i,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, 180971);