#5020791

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2478 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)(11,15)(14,16)(17,18,23,26)(19,27,20,24)(21,22,32,25)(28,31,30,29);(10,13)(14,16)(17,19,23,20)(18,24,26,27)(21,28,32,30)(22,29,25,31);(6,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,24,26,27)(21,30,32,28)(22,29,25,31);(1,2)(3,4)(5,7,8,6)(17,21)(18,25)(19,28)(20,30)(22,26)(23,32)(24,31)(27,29);(9,11)(10,14)(12,15)(13,16)(17,22,23,25)(18,21,26,32)(19,29,20,31)(24,28,27,30)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=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, 4927987);