#5894488

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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=hi,d^2=i,e^2=fg,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, 5801684);