#4428877

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1809 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:

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

Polycyclic presentation:

2^9|b^2=fgi,c^2=fh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4336073);