#263578

order	512
nilpotency class	3
isoclinism	isoclinic class of #263555
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

740 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)(3,7)(4,14)(5,17)(6,10)(8,16)(9,20)(11,28)(12,26)(13,23)(15,27)(18,22)(19,25)(21,30)(24,31)(29,32)(33,34,40,44)(35,38,43,39)(36,45,51,60)(37,46,55,58)(41,50,47,49)(42,54,59,53)(48,57,52,56)(61,64,62,63);(1,3,6,13)(2,7,10,23)(4,11,15,29)(5,18,20,12)(8,24,25,21)(9,26,17,22)(14,28,27,32)(16,31,19,30)(33,35,38,44,40,43,39,34)(36,47,49,60,51,41,50,45)(37,46,53,59,55,58,54,42)(48,57,63,62,52,56,64,61);(1,4,6,15)(2,8,10,25)(3,11,13,29)(5,19,20,16)(7,21,23,24)(9,27,17,14)(12,31,18,30)(22,32,26,28)(33,36,40,51)(34,41,44,47)(35,45,43,60)(37,52,55,48)(38,49,39,50)(42,57,59,56)(46,62,58,61)(53,64,54,63);(4,15)(8,25)(11,29)(14,27)(16,19)(21,24)(28,32)(30,31)(33,37,39,54,40,55,38,53)(34,42,43,58,44,59,35,46)(36,48,50,64,51,52,49,63)(41,56,60,62,47,57,45,61)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gh,d^c=di,f^c=fi,g^c=gh

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, 170774);