#4730191

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1665 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(18,21)(19,20)(25,26,29,31)(27,28,32,30);(2,6)(3,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,21)(19,23,20,24)(25,27)(26,30)(28,31)(29,32);(2,6)(4,7)(11,16)(13,15)(17,19,22,20)(18,23,21,24)(25,27,29,32)(26,30,31,28);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,19)(18,23,21,24)(25,28)(26,27)(29,30)(31,32);(1,4,5,7)(2,3,6,8)(9,11)(10,13)(12,16)(14,15)(17,21,22,18)(19,23,20,24)(25,27)(26,28)(29,32)(30,31)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fghi,c^2=fgi,d^2=fghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4637387);