#4651139

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

2551 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,23,24,21)(25,26,29,27)(28,30,32,31);(1,3,5,8)(2,4,6,7)(11,16)(13,15)(17,19)(18,21)(20,24)(22,23)(26,27)(30,31);(2,6)(3,8)(9,11)(10,13)(12,16)(14,15)(17,18,20,22)(19,21,24,23)(25,27,29,26)(28,30,32,31);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,24)(22,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fhi,c^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4558335);