#4379627

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	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2896 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,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, 4286823);