#5977930

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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=h,d^2=g,e^2=gh,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, 5885126);