#4300086

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fhi,c^2=gh,d^2=fgh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

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, 4207282);