#3335265

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2634 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3242461);