#3464124

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

2404 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)(11,12,15,14)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31);(6,8)(9,11,13,15)(10,14,16,12)(19,26)(22,25)(27,32)(30,31);(5,6)(7,8)(9,10,13,16)(11,14,15,12)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32);(1,2,3,4)(9,11)(10,14)(12,16)(13,15)(17,19,21,26)(18,22,24,25)(20,27,29,32)(23,30,28,31);(9,12,13,14)(10,15,16,11)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=f,d^2=fi,e^2=fg,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, 3371320);