#3974306

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2035 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3881502);