#4001500

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

2718 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fhi,d^2=fhi,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, 3908696);