#5884114

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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

1316 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)(3,7,8,6)(9,10,13,11)(12,15,16,14)(18,22)(19,25)(20,28)(30,32)(36,43)(39,46)(41,47)(44,48);(1,3)(2,6)(4,7)(5,8)(18,22)(20,28)(23,26)(27,31)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(3,8)(6,7)(9,11,13,10)(12,15,16,14)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,3)(2,7)(4,6)(5,8)(9,12,13,16)(10,14,11,15)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(1,4,5,2)(3,7,8,6)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36,37,43)(34,39,40,46)(35,41,42,47)(38,44,45,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgh,c^2=gi,d^2=fghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=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, 5791310);