#3228918

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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

2401 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=gh,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,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, 3136114);