#5421352

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,d^2=fgh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=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, 5328548);