#4435093

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

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

3045 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=i,e^2=fh,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, 4342289);