#1845727

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

1304 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=gh,e^2=g,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, 1752923);