#3547152

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2099 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=i,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3454348);