#3465339

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2597 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)(10,12)(11,15)(17,18)(19,27)(20,30)(21,29)(22,23)(24,28)(25,31)(26,32);(5,6,7,8)(11,15)(14,16)(18,23)(19,28)(25,30)(26,29);(5,6)(7,8)(9,10,13,12)(11,14,15,16)(17,19)(18,24)(20,29)(21,25)(22,28)(23,27)(26,31)(30,32);(1,2,3,4)(9,11)(10,14)(12,16)(13,15)(17,20,22,31)(18,25,23,30)(19,29,28,26)(21,27,32,24);(9,12,13,10)(11,16,15,14)(17,21)(18,26)(19,30)(20,24)(22,32)(23,29)(25,28)(27,31)

Polycyclic presentation:

2^9|b^2=fh,c^2=fg,d^2=fgi,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, 3372535);