#4379629

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2180 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgh,c^2=g,d^2=fhi,e^2=fhi,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, 4286825);