#2031224

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

1314 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)(9,10,13,12)(11,14,16,15)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34)(35,36)(37,40)(38,39)(41,44)(42,43)(45,46)(47,48);(1,2,5,4)(3,6,8,7)(9,10,13,12)(11,15,16,14)(18,22)(19,26)(20,29)(30,31)(34,39)(35,41)(37,45)(43,48);(1,2,5,4)(3,6,8,7)(9,11,13,16)(10,14,12,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,35,38,41)(34,36,39,44)(37,42,45,47)(40,43,46,48);(1,3,5,8)(2,6,4,7)(9,12,13,10)(11,15,16,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,4,5,2)(3,7,8,6)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fghi,c^2=hi,d^2=fghi,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, 1938420);