#273254

order	512
nilpotency class	3
isoclinism	isoclinic class of #273243
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	#3522	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

767 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)(3,11,14,7)(4,16,17,8)(5,10)(6,9)(12,19,23,15)(13,22)(18,29,30,24)(20,21)(25,28)(26,31)(27,32)(33,34)(35,45,48,42)(36,41)(37,52,53,40)(38,43)(39,44)(46,56,63,61)(47,60)(49,55,62,64)(50,57)(51,58)(54,59);(1,3)(2,7)(4,13)(5,19)(6,14)(8,22)(9,11)(10,12)(15,21)(16,25)(17,28)(18,31)(20,23)(24,26)(27,30)(29,32)(33,35)(34,40)(36,46)(37,47)(38,53)(39,48)(41,55)(42,43)(44,52)(45,60)(49,50)(51,63)(54,62)(56,59)(57,61)(58,64);(1,2)(3,12)(4,8)(5,20)(6,9)(7,19)(10,21)(11,15)(13,26)(14,23)(16,17)(18,29)(22,31)(24,30)(25,27)(28,32)(33,36,39,51)(34,41,44,58)(35,46,48,63)(37,49,53,62)(38,54,47,50)(40,55,52,64)(42,56,45,61)(43,59,60,57);(1,4,6,17)(2,8,9,16)(3,13,14,28)(5,18,21,30)(7,22,11,25)(10,24,20,29)(12,26,23,32)(15,27,19,31)(33,37,39,53)(34,42,44,45)(35,47,48,38)(36,49,51,62)(40,43,52,60)(41,56,58,61)(46,50,63,54)(55,59,64,57)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,e^2=i,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,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180450);