#4929613

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1940 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,23,24,22)(27,31)(30,32);(1,2)(3,6)(4,5)(7,8)(17,19,21,24)(18,22,20,23)(25,26,29,28)(27,30,31,32);(2,5)(6,7)(9,10,12,14)(11,13,15,16)(17,19,21,24)(18,22,20,23)(25,27,29,31)(26,30,28,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,32)(29,31);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,13,14,16)(17,20,21,18)(19,23,24,22)(25,28,29,26)(27,32,31,30)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4836809);