#127900

order	512
nilpotency class	4
isoclinism	isoclinic class of #7262
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

401 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=hi,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,d^b=dh,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35096);