#6344241

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

2605 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:

(1,2,6,12,7,10,5,11)(3,13,15,26,16,8,20,22)(4,9,17,24,18,23,19,25)(14,27,29,31,30,21,28,32)(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)(2,8,10,13)(4,14,18,30)(5,15,6,20)(9,21,23,27)(11,22,12,26)(17,28,19,29)(24,31,25,32)(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)(5,19)(6,17)(7,18)(8,21)(10,23)(11,25)(12,24)(13,27)(15,29)(16,30)(20,28)(22,32)(26,31)(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);(2,10)(8,13)(9,23)(11,12)(21,27)(22,26)(24,25)(31,32)(34,41)(36,47)(39,53)(42,49)(43,56)(44,58)(48,60)(59,62);(1,5,7,6)(2,11,10,12)(3,15,16,20)(4,17,18,19)(8,22,13,26)(9,24,23,25)(14,28,30,29)(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,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, 6251437);