#5753668

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5660864);