#4375621

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

2152 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gh,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,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, 4282817);