#6370626

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2386 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=f,e^2=fg,g^2=i,b^a=bf,d^a=dfi,c^b=cfi,e^b=ei

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, 6277822);