#4380495

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4326 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)(5,6,7,8)(9,10)(11,13)(12,15)(14,16)(18,23)(19,25)(20,27)(29,32);(10,11)(14,15)(17,18,22,23)(19,21,25,30)(20,28,27,24)(26,32,31,29);(5,6)(7,8)(9,10,13,11)(12,15,16,14)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32);(1,2)(3,4)(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,20,22,27)(18,24,23,28)(19,26,25,31)(21,29,30,32);(9,12)(10,14)(11,15)(13,16)(17,21)(18,25)(19,23)(20,29)(22,30)(24,31)(26,28)(27,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=h,d^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, 4287691);