#1816182

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1628 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)(3,7)(4,5)(6,8)(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(2,5)(3,8)(9,10,13,12)(11,14,16,15)(18,20)(19,23)(26,27)(29,37)(31,33)(35,40);(2,5)(6,7)(17,18,21,20)(19,24,23,22)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(17,19,21,23)(18,22,20,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,15,16,14)(17,20,21,18)(19,24,23,22)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=h,d^2=fghi,e^2=fi,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, 1723378);