#239136

order	512
nilpotency class	3
isoclinism	isoclinic class of #233136
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1230 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)(6,10)(7,8)(11,12)(13,14)(15,23)(16,17)(18,22)(19,21)(20,25)(24,26)(27,28);(6,11)(7,8)(10,12)(13,15,19,27)(14,20,21,24)(16,23,22,28)(17,26,18,25);(2,4)(5,6)(7,11)(8,10)(9,12)(13,16,17,14,19,22,18,21)(15,24,25,28,27,20,26,23);(6,12)(10,11)(14,21)(16,22)(20,24)(23,28)

Polycyclic presentation:

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

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, 146332);