#6034613

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

1292 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)(3,6)(4,5)(7,8)(9,10,14,13)(11,16,15,12)(19,25)(23,28)(26,31)(30,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(2,4)(3,7)(10,13)(11,15)(18,22)(19,25)(24,29)(26,31)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,2,5,4)(3,6,7,8)(9,11)(10,12)(13,16)(14,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,3,5,7)(2,6,4,8)(9,12,14,16)(10,15,13,11)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,4,5,2)(3,8,7,6)(9,13,14,10)(11,16,15,12)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=fi,d^2=f,e^2=h,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, 5941809);