#4379712

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#537	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4872 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:

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

Polycyclic presentation:

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

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, 4286908);