#240355

order	512
nilpotency class	3
isoclinism	isoclinic class of #233055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1180 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147551);