#513553

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂²	C₂³×C₄²
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420749);