#4382150

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 2097152
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1994 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)(3,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(19,23)(22,24)(25,26,28,30)(27,31,32,29);(2,3)(6,7)(9,11,15,14)(10,12,16,13)(18,21)(22,24)(27,32)(29,31);(1,2,5,3)(4,7,8,6)(9,12,15,13)(10,11,16,14)(17,18)(19,22)(20,21)(23,24)(25,27,28,32)(26,29,30,31);(1,3,5,2)(4,7,8,6)(9,13,15,12)(10,11,16,14);(1,4,5,8)(2,6,3,7)(9,14,15,11)(10,12,16,13)(17,19)(18,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=g,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, 4289346);