#6020073

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

1352 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gi,d^2=fgh,e^2=f,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, 5927269);