#4030106

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,c^2=fgi,d^2=gi,e^2=fg,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, 3937302);