#1907210

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

1561 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=i,e^2=fhi,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, 1814406);