#5034876

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 6291456
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2629 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef

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, 4942072);