#466713

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fi,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=de,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373909);