#5992118

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 1048576
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1729 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fgh,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5899314);