#6271071

order	512
nilpotency class	2
isoclinism	isoclinic class of #6132643
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

1350 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ghi,c^2=ghi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef

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, 6178267);