#5602140

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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

1302 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)(10,14)(11,15)(20,27)(24,29)(25,31)(30,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(9,10,13,14)(11,16,15,12)(18,22)(23,28)(24,29)(30,32)(34,38)(36,44)(39,41)(42,48);(9,11,13,15)(10,12,14,16)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,34)(35,39)(36,45)(37,38)(40,44)(41,43)(42,47)(46,48);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,3,4,7)(2,5,6,8)(9,12)(10,11)(13,16)(14,15)(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fgh,d^2=gh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,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, 5509336);