#273270

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	#3520	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

778 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)(3,10,5,8)(4,9)(6,12)(7,11)(13,22,16,25)(14,28,19,23)(15,24,31,32)(17,27)(18,26)(20,30)(21,29)(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);(1,3,7,5)(2,8,11,10)(4,15,17,31)(6,19,21,14)(9,24,27,32)(12,28,29,23)(13,20,16,18)(22,30,25,26)(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);(1,4)(2,9)(3,13)(5,16)(6,20)(7,17)(8,22)(10,25)(11,27)(12,30)(14,31)(15,19)(18,21)(23,32)(24,28)(26,29)(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);(1,5,7,3)(2,10,11,8)(4,16,17,13)(6,19,21,14)(9,25,27,22)(12,28,29,23)(15,18,31,20)(24,26,32,30)(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,b^2=h,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, 180466);