#6344253

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343554
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
prime radical	#62838	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=hi,f^2=i,g^2=hi,b^a=bf,d^a=dh,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251449);