#233351

order	512
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1790 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)(7,9)(8,11)(10,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,28)(26,27);(2,4)(5,6)(7,9)(8,11)(10,12)(13,15)(14,19)(16,24)(17,25)(18,26)(20,23)(21,27)(22,28);(6,9)(7,12)(10,11)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,28)(26,27);(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,23,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140547);