#4423580

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=g,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, 4330776);