#264609

order	512
nilpotency class	3
isoclinism	isoclinic class of #264555
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi,f^c=fi,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, 171805);