#6347227

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×#538	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,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, 6254423);