#6347225

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347203
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2589 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)(3,11)(4,8)(5,17)(6,9)(7,13)(10,19)(12,25)(14,24)(15,22)(16,29)(18,21)(20,27)(23,28)(26,32)(30,31)(33,34,38,42)(35,39,45,52)(36,40,47,54)(37,48,49,41)(43,55,57,50)(44,58,59,51)(46,60,61,53)(56,62,64,63);(1,3,6,14)(2,7,9,21)(4,12,15,26)(5,18,19,13)(8,20,22,31)(10,24,17,11)(16,30,28,27)(23,32,29,25)(33,35)(34,39)(36,43)(37,44)(38,45)(40,50)(41,51)(42,52)(46,56)(47,57)(48,58)(49,59)(53,62)(54,55)(60,63)(61,64);(1,4,6,15)(2,8,9,22)(3,12,14,26)(5,16,19,28)(7,20,21,31)(10,23,17,29)(11,25,24,32)(13,27,18,30)(33,36)(34,40)(35,43)(37,46)(38,47)(39,50)(41,53)(42,54)(44,56)(45,57)(48,60)(49,61)(51,62)(52,55)(58,63)(59,64);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)(33,37,38,49)(34,41,42,48)(35,44,45,59)(36,46,47,61)(39,51,52,58)(40,53,54,60)(43,56,57,64)(50,62,55,63);(2,9)(7,21)(8,22)(10,17)(11,24)(20,31)(23,29)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254421);