#5797048

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1825 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5704244);