#4063973

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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

2092 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3971169);