#4457464

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3632 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)(9,10,13,11)(12,15,16,14)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31);(6,8)(9,10)(11,13)(12,14)(15,16)(20,29)(24,28)(26,32)(30,31);(6,8)(10,11)(14,15)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31);(1,2,3,4)(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31);(9,12)(10,14)(11,15)(13,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=g,c^2=fg,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4364660);