#262449

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

926 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169645);