#6034430

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,5)(6,8)(11,15)(14,16)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(2,5)(3,8)(10,13)(14,16)(20,27)(23,25)(28,32)(30,31)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,2)(3,6)(4,5)(7,8)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

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, 5941626);