#6344239

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:

(1,2,6,10,7,11,5,19)(3,12,14,22,15,8,20,23)(4,9,18,24,17,25,16,29)(13,26,27,31,28,21,30,32)(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);(1,3,7,15)(2,8,11,12)(4,13,17,28)(5,14,6,20)(9,21,25,26)(10,22,19,23)(16,27,18,30)(24,31,29,32)(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)(2,9)(3,13)(5,16)(6,18)(7,17)(8,21)(10,24)(11,25)(12,26)(14,27)(15,28)(19,29)(20,30)(22,31)(23,32)(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);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,16,17,18)(8,22,12,23)(9,24,25,29)(13,27,28,30)(21,31,26,32)(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);(4,17)(9,25)(13,28)(16,18)(21,26)(24,29)(27,30)(31,32)(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=h,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, 6251435);