#4343027

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2539	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 4250223);