#1784648

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

1306 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)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34,37,41)(35,39,43,38)(36,40,45,48)(42,47,44,46);(10,13)(14,16)(18,22)(19,26)(20,29)(30,31)(33,35,37,43)(34,38,41,39)(36,44,45,42)(40,47,48,46);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,45)(46,48);(1,3,5,8)(2,6,4,7)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,47)(45,48);(1,4,5,2)(3,7,8,6)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=hi,e^2=fgi,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, 1691844);