#4129436

order	512
nilpotency class	2
isoclinism	isoclinic class of #4122169
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4036632);