#4456247

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2528 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)(3,4)(6,8)(11,15)(14,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29);(2,4)(5,6,7,8)(10,13)(14,16)(19,26)(22,25)(27,32)(29,31);(2,4)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,4)(5,6)(7,8)(17,19,21,26)(18,22,24,25)(20,27,28,32)(23,29,30,31);(9,11)(10,14)(12,15)(13,16)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgi,c^2=fgi,d^2=fg,e^2=fg,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, 4363443);