#4428874

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

1915 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=h,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, 4336070);