#5022374

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
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	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2634 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)(11,12)(13,15)(14,16)(17,18,21,24)(19,23,20,22)(25,26)(27,29)(28,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,15,12)(17,19,21,20)(18,22,24,23)(25,27,30,31)(26,28,32,29);(2,5)(3,7)(10,15)(12,14)(17,19,21,20)(18,23,24,22)(25,28,30,29)(26,31,32,27);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20,21,19)(18,23,24,22)(25,29,30,28)(26,31,32,27);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,16,15,11)(17,18)(19,22)(20,23)(21,24)(25,26)(27,28)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fg,d^2=fg,e^2=g,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, 4929570);