#4940794

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3648 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)(3,4)(5,6,7,8)(11,15)(14,16)(18,22)(23,29)(24,27)(30,31);(10,13)(14,16)(17,18,21,22)(19,23,26,29)(20,27,28,24)(25,31,32,30);(2,3)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32);(1,3,4,2)(5,6)(7,8)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(9,11)(10,14)(12,15)(13,16)(17,20,21,28)(18,24,22,27)(19,25,26,32)(23,30,29,31)

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4847990);