#5164279

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1318 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=fg,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 5071475);