#4456265

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2060 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=fi,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, 4363461);