#1900869

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
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

1298 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)(18,22)(20,27)(23,28)(26,31)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,47)(46,48);(1,2,5,4)(3,6,8,7)(9,10,13,12)(11,15,16,14)(18,22)(19,25)(24,29)(26,31)(34,37)(35,42)(36,45)(47,48);(1,2,5,4)(3,6,8,7)(9,11,13,16)(10,14,12,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35)(34,39)(36,46)(37,41)(38,42)(40,48)(43,45)(44,47);(1,3,5,8)(2,6,4,7)(9,12,13,10)(11,15,16,14)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,47)(41,48)(42,46);(1,4,5,2)(3,7,8,6)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,34)(35,41,42,39)(36,44,45,40)(43,48,46,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=fghi,c^2=fghi,d^2=hi,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1808065);