#284238

order	512
nilpotency class	3
isoclinism	isoclinic class of #264491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

773 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:

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

Polycyclic presentation:

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

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, 191434);