#3335220

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

2673 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)(10,11)(12,16)(17,18,22,19)(20,25,27,23)(21,24,30,26)(28,32,29,31);(5,6,7,8)(10,11)(14,15)(18,19)(20,27)(24,26)(28,29);(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,19,22,18)(20,25,27,23)(21,24,30,26)(28,31,29,32);(1,2)(3,4)(9,11,13,10)(12,15,16,14)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30);(9,12)(10,14)(11,15)(13,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=fg,d^2=g,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, 3242416);