#4331408

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1993 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fi,d^2=f,e^2=fh,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, 4238604);