#1848747

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

1682 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,21)(22,24)(26,31)(28,35)(33,37)(36,40);(2,3)(4,8)(18,20)(19,24)(25,26,30,31)(27,28,32,35)(29,37,38,33)(34,40,39,36);(2,3)(6,7)(17,18,21,20)(19,23,24,22)(25,27)(26,28)(29,34)(30,32)(31,35)(33,36)(37,40)(38,39);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,37)(38,40);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,20,21,18)(19,23,24,22)(25,29,30,38)(26,33,31,37)(27,34,32,39)(28,36,35,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=h,d^2=g,e^2=ghi,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, 1755943);