#5482222

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1394 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)(11,15)(13,16)(18,22)(20,28)(23,25)(26,32)(33,34,38,39)(35,40,42,36)(37,41,45,47)(43,48,46,44);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(18,22)(20,28)(23,25)(26,32)(34,39)(35,42)(37,45)(44,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35)(34,40)(36,39)(37,46)(38,42)(41,44)(43,45)(47,48);(9,11)(10,13)(12,15)(14,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,40)(34,35,39,42)(37,44,45,48)(41,43,47,46);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,41,39,47)(35,43,42,46)(36,44,40,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=fg,e^2=f,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, 5389418);