#6252679

order	512
nilpotency class	2
isoclinism	isoclinic class of #6132643
automorphism	Aut(G) order 5242880
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1251 conjugacy classes of subgroups; only the 440 normal subgroups are shown.

normal subgroups only

This group has 1251 conjugacy classes of subgroups — too many to draw. Only its 440 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=fghi,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef

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, 6159875);