#6344246

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	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2606 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=i,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, 6251442);