#2046832

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1722 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fgh,d^2=fh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1954028);