#148706

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=d,e^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^c=eh,h^c=hi

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, 55902);