#3547145

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2252 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=fgi,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3454341);