#4314455

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
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

3584 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)(9,10,13,14)(11,15,16,12)(19,25)(20,27)(22,28)(23,30);(10,14)(12,15)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32);(5,6)(7,8)(9,11,13,16)(10,15,14,12)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32);(9,12)(10,11)(13,15)(14,16)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^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, 4221651);