#6386919

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂⁴×C₈
socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2434 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)(5,6,7,9,8,10,11,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,28)(25,27);(13,15)(14,19)(16,23)(17,25)(18,24)(20,27)(21,22)(26,28);(1,2,3,4)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,22,28);(15,24)(19,22)(23,25)(27,28);(13,17,18,16)(14,20,21,26)(15,25,24,23)(19,27,22,28)

Polycyclic presentation:

2^9|a^2=h,c^2=fg,e^2=f,h^2=i,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, 6294115);