#4380483

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3514 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)(9,10,13,11)(12,15,16,14)(17,18,23,21)(19,27,29,24)(20,25,28,30)(22,26,32,31);(10,11)(14,15)(17,19,23,29)(18,24,21,27)(20,22,28,32)(25,31,30,26);(5,6)(7,8)(9,10,13,11)(12,15,16,14)(17,20)(18,25)(19,22)(21,30)(23,28)(24,31)(26,27)(29,32);(1,2)(3,4)(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,21,23,18)(19,24,29,27)(20,30,28,25)(22,31,32,26);(9,12)(10,14)(11,15)(13,16)(17,22)(18,26)(19,28)(20,29)(21,31)(23,32)(24,25)(27,30)

Polycyclic presentation:

2^9|a^2=f,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, 4287679);