#1982841

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fg,c^2=fghi,d^2=fghi,e^2=hi,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, 1890037);