#284345

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

769 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fh,c^2=hi,d^2=i,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, 191541);