#4032890

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2618 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)(3,5)(4,7)(6,8)(9,10,14,16)(11,13,12,15)(18,22)(20,23)(25,26)(27,29)(28,31)(30,32);(2,3)(4,8)(9,11,14,12)(10,15,16,13)(17,18)(19,23)(20,24)(21,22)(25,26,28,31)(27,29,32,30);(4,8)(6,7)(10,16)(13,15)(18,22)(20,23)(25,27,28,32)(26,29,31,30);(1,3,5,2)(4,6,8,7)(9,12,14,11)(10,13,16,15)(17,19,21,24)(18,20,22,23)(25,26)(27,29)(28,31)(30,32);(1,4)(2,6)(3,7)(5,8)(9,13)(10,12)(11,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=fghi,c^2=fhi,d^2=hi,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, 3940086);