#4456272

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2179 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)(3,8)(4,6)(5,7)(9,10)(11,12)(13,16)(14,15)(18,23)(19,21)(27,31)(28,32);(1,3,6,5)(2,7,4,8)(10,15)(12,16)(17,18,22,23)(19,20,21,24)(25,26,29,30)(27,32,31,28);(2,4)(7,8)(9,11,14,13)(10,16,15,12)(17,19,22,21)(18,24,23,20)(25,27)(26,28)(29,31)(30,32);(1,4,6,2)(3,7,5,8)(9,12,14,16)(10,11,15,13)(17,20,22,24)(18,19,23,21)(25,26)(27,28)(29,30)(31,32);(1,5,6,3)(2,7,4,8)(9,13,14,11)(10,16,15,12)(17,21,22,19)(18,24,23,20)(25,28)(26,27)(29,32)(30,31)

Polycyclic presentation:

2^9|b^2=hi,c^2=fi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df

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, 4363468);