#127480

order	512
nilpotency class	4
isoclinism	isoclinic class of #127215
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

531 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,9,6,10,5,8)(3,11,13,7,12,16,14,15)(17,18,22,27,23,28,20,25)(19,29,30,44,32,45,33,48)(21,26,34,40,36,42,35,41)(24,37,38,31,39,46,43,47);(1,3,5,14,6,12,4,13)(2,7,9,16,10,15,8,11)(17,19)(18,24,28,39)(20,33)(21,31)(22,30)(23,32)(25,38,27,43)(26,29,42,45)(34,46)(35,37)(36,47)(40,48,41,44);(3,12)(7,15)(11,16)(13,14)(17,20,23,22)(18,25,28,27)(19,30,32,33)(21,35,36,34)(24,38,39,43)(26,41,42,40)(29,44,45,48)(31,46,47,37)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,e^2=gh,f^2=ghi,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cg,e^b=eh,d^c=di,e^d=ei

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, 34676);