#5035248

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3422 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=i,e^2=f,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, 4942444);