#4929361

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
automorphism	Aut(G) order 2097152
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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2702 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4836557);