#134600

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,e^a=eg,e^c=eh,f^c=fgh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41796);