#4942024

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 4194304
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	C₂²×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3836 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)(9,10,12,14)(11,13,15,16)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32);(11,15)(13,16)(18,22)(20,27)(23,28)(25,31);(2,3)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32);(1,3,4,2)(5,6)(7,8)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32);(9,10)(11,13)(12,14)(15,16)(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)

Polycyclic presentation:

2^9|a^2=fghi,c^2=fh,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, 4849220);